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0, and R for i = 0. The tensor powers form a graded R -module .
.
1 NM = M M i : i=0
The assignment ((x1 . . . xm ) (y1 . . . yn)) 7;! x1 xm y1 yn m M n ! M (m+n), and its additive induces an RN-bilinear map MN N extension to M M gives M the structure of a graded associative R -algebra. Henceforth `R -algebra' always means `associative R -algebra'. N (Obviously M is not commutative in general.) The tensor algebra is characterized by a universal property: given an R -linear map ' : M ! A where N A is an R-algebra, there exists a unique R-algebra homomorphism : M ! A extendingV' here we identify M and M 1 . The exterior algebra M is the residue class algebra ^ N M = ( M )= where is the two-sided ideal generated by theV elements x x, x 2 M . Since is generated by homogeneous elements, M inherits the structure V M is denoted of a graded R -algebra. The product in V M is not commutative it is however alternating : one xhas^ y. In general ^ for homogeneous x y 2 M and x ^ y = (;1)(deg x)(deg y) y ^ x x^x= 0 for homogeneous x deg x odd: J
J
J
41
1.6. The Koszul complex
Let x1 . . . xn be elements of M , and a permutation of f1 . . . ng. Then x(1) ^ ^ x(n) = ( )x1 ^ ^ xn here ( ) is the sign of . Furthermore x1 ^ ^ xn = 0 if xi = xj for some indices i 6= j . For a subset I of f1 . . . ng we set xI = xi ^ ^ xim when I = fi1 . . . im g with i1 < < im : For subsets J , K f1 . . . ng with J \ K = let (J K ) = (;1)i where i is the number of elements (j k) 2 J K with j > k if J \ K 6= , let (J K ) = 0. Then xJ ^ xK = (J K )xJ K : Useful identities satis ed by are given in Exercise 1.6.23. It is clear that the notation xI can be extended to the more general case in which (xg )g2G is a family of elements of M indexed by a linearly ordered set G and I is a nite subset of G. V V The i-th graded component of M is denoted byV i M and is called the i-th exterior power of M . From the de nitionVof M it follows easily V 0 that one has natural isomorphisms M = R, 1 M = M so we may V V 0 1 identify R and M , M and M . V Let (xg )g2G be a system of generators of M . Then j M is generated by the exterior products xI with V I G and jI j = j. In particular, if M is generated by x1 . . . xn, then i M = 0 for all i > n. The exterior algebra is characterized by a universal property which it inherits from that of the tensor algebra: given an R -linear map ' : M ! E from M to an R -algebra E such that '(xV)2 = 0 for all x 2 M , there exists a unique R -algebra homomorphism : M ! E extending '. It follows immediately that for every RV-linear map ' : M ! N there exists a unique R -algebra homomorphism ' for which the diagram 1
' ;;;;! N ?? ??
M
y V ynat ' V M ;;;;! VN
nat
V
commutes ' is homogeneous of degree 0, and one has
^
'(x1
^ ^ xn) = '(x1) ^ ^ '(xn) V
for all and V x1 . . . xn 2 M. If ' is surjective, then ' is also surjective, Ker ' is the ideal generated by Ker '. (This is neither obvious V ' need notnor indeed true in general for example, if ' is injective, V V V V be injective.) The map i M ! i N Vinduced by ' is denoted by i '. Suppose that ' is surjective then i ' is also surjective, and from the
42
1. Regular sequences and depth
V
description of Ker ' just mentioned (and the alternating property of V M) it follows easily that the sequence i;1 ^ ^i Vi ' ^i M Ker ' ;! M ;;! N ;! 0 is exact where Vthe map on the leftVhand side is induced by the exterior multiplication i;1 M VKer ' ! i M . The exterior powers i M are also characterized by a universal property: for every alternating i-linear map : M i ! N , N an R -module, V i there exists a unique R -linear map : M ! N such that (x1 . . . xi ) = (x1 ^ ^ xi ) for all x1 . . . xi 2 M . An important property of the exterior algebra is that it commutes with base extensions: if R ! S is a homomorphism of commutative rings, then one has a natural isomorphism
^
( M ) R S =
^
(M R S )
of graded S -algebras. V V Let M1, M2 be R -modules. On ( M1) ( M2) one de nes a multiplication by setting (x y)(x0 y0 ) = (;1)(deg y)(deg x0 )(x ^ x0 ) (y ^ y0 ) for all homogeneous elements x, x0 2 M1, y, y0 2 M2. It is straightforward V V to verify that ( M1 ) ( M2) is an alternating graded R -algebra under this multiplication. Its degree 1 component is (M1 R ) (R M2 ) = M1M2 . By the universal property of the exterior algebra the natural map V (V M2) extends to an R-algebra homomorphism M1 M2 ! ( M1 ) V V V : (M1 M2 ) ! ( M1 ) V ( M2). V V One gets an inverse : ( M1) ( M2) ! (M1 M2) to by setting (x y) = 1 (x) ^ 2(y) V V where i : Mi ! (M1 M2 ) is the extension of the natural embedding M and are the identities on Vi ! M1 VM2. The compositions V ( M1) ( M2 ) and (M1 M2). Therefore we have an isomorphism
^
^
( M1) ( M2) =
^
(M1 M2 )
of alternating graded R -algebras. In what follows, the most important case for M is that of a nite free R -module F . Suppose e1 . . . en is a basis of F . The elements eI I f1 . . . ng jI j = i
43
1.6. The Koszul complex
V
form a basis of i F this non-trivial existence of ;nto. Athemultiplication Vi F is freefactofamounts determinants. In particular rank i V table of F with respect to this basis in given by
^ eJ = (I J )eIJ :
eI
L
Suppose R is a graded V ring, and M = i2Z Mi is a graded R-module. V Then one can endow M Vwith a unique grading such that M M has the given grading, and MLis a graded algebra over R . We restrict ourselves to the case M = F = ni=1 R (;ai). Let e1 . . . en be the basis of F corresponding to this decomposition. Then one assigns eI the degree P V toF makes VF a a , and veri es easily that the induced grading on i2I i graded (in fact, a bigraded) R -algebra. Basic properties of the Koszul complex. Let R be a ring, L an R -module, and f : L R an R -linear map. The assignment
!
(x1 . . . xn) 7!
Xn i=1
(;1)i+1f (xi )x1 ^ ^ bxi ^ ^ xn
V
de nes an alternating n-linear map Ln ! n;1 L. (By bxi we indicate that xi is to be omitted from the exterior product.) By the universal V property V of the n-th exterior power there exists an R -linear map df(n) : n L ! n;1 L with df(n)(x1
^ ^ xn) = X(;1)i+1f(xi)x1 ^ ^ bxi ^ ^ xn n
i=1
for all x1 . . . xn 2 L. The collection of the maps df(n) de nes a graded R -homomorphism ^ ^ df : L ! L
of degree ;1. By a straightforward calculation one veri es the following identities:
^
^
;
^
df df = 0 and df (x y) = df (x) y + ( 1)deg x x df (y)
V for all homogeneous x 2 L. To say that df df = 0 is to say that ^ d f d ^ ;! ^ L ;! L ;! ;! L ;! L ;! R ;! 0 n
f
n;1
2
f
is a complex. The second equation expresses that df is an antiderivation (of degree ;1).
44
1. Regular sequences and depth
De nition 1.6.1. The complex above is the Koszul complex of f , denoted by K (f ). More generally, if M is an R -module, then K (f M ) is the complex K (f ) R M , called the Koszul complex of f with coecients in M its di erential is denoted by dfM . Proposition 1.6.2. Let R be a ring, L an R-module, and f : L ! R an .
.
.
R-linear map. (a) The Koszul complex K. (f ) carriesVthe structure of an associative graded alternating algebra, namely that of L. (b) Its dierential df is an antiderivation of degree 1. (c) For every R-module M the complex K. (f M ) is a K. (f )-module in a natural way. (d) One has dfM (x : y) = df (x) : y + ( 1)deg x x : dfM (y) for all homogeneous elements x of K. (f ) and all elements y K. (f M ).
;
;
2
(a) and (b) are part of the discussion preceding the proposition. (c) is obvious: if A is an R -algebra, then A R M is an A-module for every R -module M . (d) It is enough to verify the equation for elements y = w z with w 2 K (f ), z 2 M . Then dfM (x : w z ) = dfM ((x ^ w) z ) = df (x ^ w) z , and the rest follows from the fact that df is an antiderivation. For a subset S of K (f ) and a subset U of K (f M ) let S : U denote the R -submodule of K (f M ) generated by the products s : u, s 2 S , u 2 U . Set Z (f ) = Ker df Z (f M ) = Ker dfM B (f ) = Im df B (f M ) = Im dfM : De nition 1.6.3. The homology H (f ) = Z (f )=B (f ) is the Koszul homology of f . For every R -module M the homology Z (f M )=B (f M ) is denoted by H (f M ) and called the Koszul homology of f with coecients in M . From 1.6.2(d) one easily derives the following relations: Z (f ) : Z (f M ) Z (f M ) Z (f ) : B (f M ) B (f M ) B (f ) : Z (f M ) B (f M ): We have a natural isomorphism K (f ) = K (f R ). So the rst relation entails that Z (f ) is a graded R -subalgebra of K (f ), and the second and third show that B (f ) is a two-sided ideal in Z (f ). Proposition 1.6.4. Let R be a ring, L an R-module, and f : L ! R an Proof.
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R-linear map. (a) The Koszul homology H. (f ) carries the structure of an associative graded alternating R-algebra. (b) For every R-module M the homology H. (f M ) is an H. (f )-module in a natural way.
45
1.6. The Koszul complex
(a) That H (f ) is an R -algebra follows from the discussion preceding the proposition. The asserted properties are inherited by quotients of graded R -subalgebras of K (f ) modulo graded ideals. (b) The rst of the relations above shows that Z (f M ) is a Z (f )module the second says that B (f M ) is a Z (f )-submodule, and the third implies that Z (f M )=B (f M ) is annihilated by B (f ). It results immediately from 1.6.4 that H (f M ) is an R=I -module where I = Im f . This will be stated in 1.6.5 where it follows from a somewhat stronger statement. It is useful also to introduce the Koszul cohomology (with coecients in M ): we set K (f ) = HomR (K (f ) R ) K (f M ) = HomR (K (f ) M ) H (f ) = H (K (f )) H (f M ) = H (K (f M )): Proof.
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Let I = Im f R then, by construction, H0 (f ) = R=I and H0 (f M ) = M=IM . Proposition 1.6.5. Let R be a ring, L an R-module, and f : L ! R an R-linear map. Set I = Im f. (a) For every a 2 I multiplication by a on K (f ), K (f M ), K (f ), K (f M ) .
.
.
.
is null-homotopic. (b) In particular I annihilates H. (f ), H. (f M ), H . (f ), H . (f M ). (c) If I = R, then the complexes K. (f ), K. (f M ), K . (f ), K . (f M ) are null-homotopic. In particular their (co)homology vanishes.
We choose x 2 L with a = f (x). Let #a denote the multiplication by a on K (f ), and x the left multiplication by x on K (f ). Then #a = df x + x df as is easily veri ed. Thus multiplication by a is null-homotopic on K (f ). Of course #a M and HomR (#a M ) are the multiplications by a on K (f M ) and K (f M ), and the rest of (a) follows immediately. Part (b) is a general fact: if ' is a null-homotopic complex homomorphism, then the map induced by ' on homology is zero. For (c) we choose a = 1, and apply (a) and (b). Let L1 and L2 be R -modules, and f1 : L1 ! R , f2 : L2 ! R be R linear maps. Then f1 and f2 induce a linear form f : L1 L2 ! R by f (x1 x2) = f1 (x1) + f2(x2 ). Proposition 1.6.6. With the notation just introduced, one has an isomorphism of complexes K (f1) R K (f2) = K (f ). Proof. The f ), V gradedV R-algebras V underlying K (f1) K (f2) and K (We namely ( L1) ( L2 ) and L, are isomorphic, as noted above. V may identify them. The di erential df is an antiderivation on L which on the degree 1 graded component L = L1 L2 coincides with df df . Proof.
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1
2
46
1. Regular sequences and depth
V
An antiderivation on the exterior algebra L is uniquely determined by its values on L. Therefore it is enough to check that df df is an antiderivation too. The straightforward veri cation of this fact is left as an exercise for the reader. One has of course to remember the de nition of tensorLproduct the n-th graded component of V of complexes: V K (f1) K (f2) is ni=0 i L1 n;i L2 , and 1
.
2
.
df (x y) = df (x) y + (;1)ix df (y) V V for x y 2 i L1 n;i L2. df
1
2
1
2
The Koszul complex `commutes' with ring extensions, and so does Koszul homology if the extension is at: Proposition 1.6.7. Let R be a ring, L an R-module, and f : L ! R an R-linear map. Suppose ' : R ! S is a ring homomorphism. (a) Then one has a natural isomorphism K (f ) R S = K (f S ). (b) Moreover, if ' is at, then H (f M ) S = H (f S M S ) for every .
.
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R-module M.
.
V
V
There is a natural isomorphism ( L) S = (L S ), and df S and dfS are antiderivations which coincide in degree 1. So we can use the same argument as in the previous demonstration. This proves (a), and (b) follows immediately since H (C S ) = H (C ) S for an arbitrary complex C over R if S is R -at. Suppose L and L0 are R -modules with linear forms f : L ! R and 0 f : L0 ! R . Every RV -homomorphism ' : L ! L0 extends to a homomorV V phismV ' : L ! L0 of R -algebras, as discussed above. If f = f 0 ', then ' is a homomorphism of Koszul complexes: Proposition With the notation just introduced, if f = f 0 ', then V ' : K (f) !1.6.8. K (f 0 ) is a complex homomorphism. Proof.
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The Koszul complex of a sequence. Let L be a nite free R -module with basis e1 . . . en . Then a linear form f on L is uniquely determined by the values xi = f (ei ), i = 1 . . . n. Conversely, given a sequence x = x1 . . . xn, there exists a linear form f on L with f (ei) = xi . We set K. (x) = K. (f )
and the rest of the notation is to be modi ed accordingly. Henceforth we shall only consider Koszul complexes K (x). Since f is just the `direct sum' of the linear forms fi : R ! R , fi (1) = xi , 1.6.6 specializes to the isomorphism .
K. (x) = K. (x0)
K (xn) = K (x1) K (xn) .
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47
1.6. The Koszul complex
where x0 = x1 . . . xn;1. Furthermore one should note that, by 1.6.8, K (x) is essentially invariant under a permutation of x. We set I = (x). Let F be a free resolution of R=I . As H0 (x) = R=I , there exists a complex homomorphism ' : K (x) ! F lifting the identity on R=I note that ' is unique up to homotopy. .
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Proposition 1.6.9. Let R be a ring, x = x1 . . . xn a sequence in R, and I = (x). For all i there exist natural homomorphisms Hi (x M ) Proof.
! ToriR (R=I M)
and ExtiR (R=I M )
! H i(x M):
The map ' introduced above yields complex homomorphisms ! F M and HomR (' M): HomR (F M) ! K (f M).
' M : K. (f M )
.
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.
Let L be V a nite free R-module with basis e1 . . . en. Then! e:1 ^^ Vn L !en Ris a basis of n L, and there exists a unique R -isomorphism n Vn L = R is usually called with !n (e1 ^ ^ en ) = 1. (An isomorphism V V an orientation on L.) We de ne !i : i L ! ( n;i L) by setting (!i (x))(y) = !n(x ^ y)
for x 2
^i
L y
2 ^ L: n;i
(This causes no ambiguity for i = n if we identify R and R under the natural isomorphism.) It follows immediately that
0
I \ J 6= , (!i (eI ))(eJ ) = (I J ) for for I \ J = .
In this formula I and J are multi-indices as introduced above. It shows that !i is an isomorphism. If we denote the dual basis of (eI ) by (eI ), the formula says that !i (eI ) = (I I )eI where I = f1 . . . ng n I . Thus !i is an isomorphism. We consider the diagram K. (x): 0 K . (x): 0
;! ^?L ;!d ^?L ;!d ;!d L? ;!d R? ;! 0 ?! ?! ?! ?! n;1
n
y
;!
R
y
n
d ;!
L
with d = dx and d = (dx ).
n;1
d d ;! ;! (
y ^
n;1
y ^
1
L)
d ;! (
n
0
L)
;! 0
48
1. Regular sequences and depth
Proposition 1.6.10. Let x = x1 . . . xn be a sequence in a ring R. (a) With the notation just introduced, one has !i;1 di = (;1)i;1dn;i+1 !i
for all i. (b) The complexes K. (x) and K . (x) = (K. (x)) are isomorphic (we say that K. (x) is self-dual). (c) More generally, for every R-module M the complexes K. (x M ) and K . (x M ) are isomorphic, and (d) Hi (x M ) = H n;i (x M ) for i = 0 . . . n.
The veri cation of (a) is left as an exercise for the reader (1.6.23 is helpful). We observed above that !i is an isomorphism so that the maps i = (;1)i(i;1)=2!i de ne an isomorphism K (x) = K (x) = (K (x)). For (c) we note that there is a natural homomorphism N M ! HomR (N M ) for all R -modules N , M . If N is nite and free, this homomorphism is an isomorphism, and it induces an isomorphism K (x) M = HomR (K (x) M ). Now one uses (b). Part (d) is a trivial consequence of (c). The reader may have noticed that for a formally correct formulation of 1.6.10(b) one would rst have to convert the cochain complex K (x) into a chain complex C (by setting Ci = K ;i (x)) and then state that K (x) = C (;n). A similar manipulation would be necessary for (c). The Koszul complex is an exact functor: Proposition 1.6.11. Let R be a ring, x = x1 . . . xn a sequence in R, and 0 ! U ! M ! N ! 0 an exact sequence of R-modules. Then the induced Proof.
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sequence
0 ;! K (x U ) ;! K (x M ) ;! K (x N ) ;! 0 .
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is an exact sequence of complexes. In particular one has a long exact sequence
;! Hi (x U) ;! Hi(x M) ;! Hi (x N) ;! Hi;1(x U) ;!
of homology modules.
The components of K (x) are free, hence at R -modules. In place of an R -module M one can more generally consider a complex C , and then de ne the Koszul homology of C to be the homology of K (x) C etc. We consider this construction only for the special case in which x = x: Proposition 1.6.12. Let R be a ring, and x 2 R. (a) For every complex C of R-modules one has an exact sequence 0 ;! C ;! C K (x) ;! C (;1) ;! 0:
Proof.
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49
1.6. The Koszul complex
(b) The induced long exact sequence of homology is
x ;! Hi (C ) ;! Hi(C K (x)) ;! Hi;1(C ) ;! Hi;1 (C ) ;! (c) Moreover, if x is C -regular, then there is an isomorphism H (C K (x)) = H (C =xC ): (According to our convention for graded modules C (;1) is just the .
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complex C with all degrees increased by 1.) x Proof. The complex K (x) is simply 0 ;! R ;! R ;! 0. The i-th graded component of K (x) C is therefore (R Ci ) (R Ci;1) = Ci Ci;1. So we have in each degree an exact sequence 0 ;! Ci ;! Ci Ci;1 ;! Ci;1 ;! 0 where and are the natural embedding and projection. If @ is the di erential of C , then the di erential d : Ci Ci;1 ! Ci;1 Ci;2 is given by the matrix @ (;1)i;1x 0 @ according to the de nition of tensor products of complexes. Now (a) is obvious. For (b) one looks up the de nition of connecting homomorphism. It is de ned by the following chain of assignments starting from z 2 Ci;1 with @(z ) = 0: .
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;7 ! (0 z) 7;d! ((;1)i;1xz 0) 7;! (;1)i;1xz: So the connecting homomorphism Hi (C (;1)) = Hi;1(C ) ! Hi;1 (C ) is multiplication by (;1)i;1x. (c) The natural maps Ci Ci;1 ! Ci ! Ci=xCi constitute a complex homomorphism C K (x) ! C =xC . We claim that the associated map of homology is an isomorphism. In fact, let z 2 Ci such that @(z ) 2 xCi;1. Then there exists z 0 2 Ci;1 with @(z ) = xz 0 , and d (z (;1)i z 0) = (0 (;1)i@(z 0 )). Next one has x@(z 0 ) = @(@(z )) = 0 so @(z 0 ) = 0 since multiplication by x is injective on C : (z (;1)i z 0 ) is a cycle mapped to the cycle z 2 Ci=xCi. That the map of homology is injective can be veri ed z
;1
;1
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similarly. Corollary 1.6.13. Let R be a ring, x = x1 . . . xn a sequence in R, and M an R-module. (a) Set x0 = x1 . . . xn;1. Then one has an exact sequence
x x ;;! Hi (x0 M ) ! Hi (x M ) ! Hi;1 (x0 M ) ;;! Hi;1 (x0 M ) ! (b) Let p n, x0 = x1 . . . xp, and x00 = xp+1 . . . xn . If x0 is weakly M-regular, then one has an isomorphism H (x M ) = H (x00 M=x0 M ). n
n
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50
1. Regular sequences and depth
Part (a) is a special case of 1.6.12(b) when we take C = K (x0 M ) and use the isomorphisms K (x0 M ) K (xn ) = K (x0 ) M K (xn ) = K (x M ): For part (b) it is enough to do the case p = 1 from which the general case follows by induction. Next we may permute x to the sequence x2 . . . xn x1, and then the assertion follows from 1.6.12(c). It is an immediate consequence of 1.6.13 that Hi (x M ) = 0 for i = n ; p + 1 . . . n if (the rst) p elements of x1 . . . xn form an M -sequence. As we shall see in 1.6.16, there is a somewhat stronger vanishing theorem. Corollary 1.6.14. Let R be a ring, x a sequence in R, and M an R-module. (a) If x is an M-sequence, then K (x M ) is acyclic. (b) If x is an R-sequence, then K (x) is a free resolution of R=(x). Remark 1.6.15. Let R be a graded ring and x = x1 . . . xn a sequence of homogeneous elements. Then x induces a linear form of degree 0 on L F = ni=1 R (; deg xi ). The Koszul complex (x) is a graded complex V F theK grading with a di erential of degree 0 if we give discussed above. V = R(; Pn deg x ). In particular one has n F i i=1 Proof.
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The main importance of the Koszul complex stems from the fact that H (x M ) measures grade(I M ) if M is a nite module over a Noetherian ring R and I = (x). This will be made precise in 1.6.17. The niteness assumption just stated will be necessary to establish the existence of an M -sequence in I from the vanishing of certain homology modules Hi (x M ). The converse holds without such an assumption: Theorem 1.6.16. Let R be a ring, x = x1 . . . xn a sequence in R, and M an R-module. If I = (x) contains a weak M-sequence y = y1 . . . ym , then Hn+1;i(x M ) = 0 for i = 1 . . . m and Hn;m (x M ) = HomR (R=I M=y M ) = ExtmR (R=I M ): The Koszul complex and grade.
.
The last isomorphism is given by Lemma 1.2.4. The remaining claims are proved by induction on m. For m = 0 we must show that Hn (x M ) = HomR (R=I M ): In fact, by 1.6.10 one has Hn (x M ) = H 0 (x M ), and the latter is naturally isomorphic with HomR (R=I M ), as follows from the exactness of R n ! R ! R=IV ! 0 and the left exactness of HomR ( M ). Explicitly, if we identify n R n M and R M = M via an orientation !n of R n, then Hn (x M ) is just the submodule fy 2 M : Iy = 0g = HomR (R=I M ) of M . Proof.
51
1.6. The Koszul complex
= M=y1M . The exact sequence Let m 1. Then we set M
y ;! 0 0 ;! M ;! M ;! M induces an exact sequence y y ) ;! Hi;1 (x M ) ;! ;! Hi(x M) ;! Hi (x M ) ;! Hi (x M 1
1
1
see 1.6.11. Since, by 1.6.5, y1 annihilates Hi (x M ) for all i, this exact sequence breaks up into exact sequences ) ;! Hi;1(x M ) ;! 0: 0 ;! Hi (x M ) ;! Hi (x M It only remains to apply the induction hypothesis. Theorem 1.6.17. Let R be a Noetherian ring, and M a nite R-module. Suppose I is an ideal in R generated by x = x1 . . . xn . (a) All the modules Hi (x M ), i = 0 . . . n, vanish if and only if M = IM. (b) Suppose that Hi (x M ) 6= 0 for some i, and let h = maxfi : Hi (x M ) 6= 0g: Then every maximal M-sequence in I has length g = n ; h in other words, grade(I M ) = n ; h. Proof. (a) The implication `)' is trivial: M = IM () H0 (x M ) = M=IM = 0. For the converse choose a prime ideal . By 1.6.7 and the atness of localization one has (Hi (x M )) = Hi (x M ) where x is considered a sequence in R on the right hand side. If I 6 , then Hi (x M ) = 0 by 1.6.5. If I , then M = 0 by Nakayama's lemma, and again we have Hi (x M ) = 0. (b) We give two proofs. (A third proof for the case M = R is indicated in Exercise 1.6.30.) (i) By part (a) we have M 6= IM . Let y be a maximal M -sequence in I then y has length g = grade(I M ). It follows immediately from 1.6.16 and 1.2.5 that Hi (x M ) = 0 for i = n ; g + 1 . . . n and Hn;g (x M ) = ExtgR (R=I M ) 6= 0. (ii) Let y be a maximal M -sequence in I , and suppose that y has length g. Then Hi (x M ) = 0 for i = n ; g + 1 . . . n by 1.6.16, and furthermore Hn;g (x M ) = HomR (R=I M=y M ). Since I consists of zerodivisors of M=y M , this module is non-zero see 1.2.3. The second proof just given is independent of the `homological' Lemma 1.2.4, and shows again that all maximal M -sequences in I have the same length. Therefore one could build the theory of grade upon 1.6.17. Corollary 1.6.14 can be reversed for local rings. We need the following lemma: p
p
p
p
p
p
p
p
p
52
1. Regular sequences and depth
Lemma 1.6.18. Let (R ) be a Noetherian local ring, M a nite R-module, and x = x1 . . . xn a sequence in . Set x0 = x1 . . . xn;1. If Hi (x M ) = 0, then Hi (x0 M ) = 0. Proof. By 1.6.6 we have K (x) = K (x0) K (xn ). So 1.6.13 gives us an exact sequence m
m
.
Hi (x0 M )
.
.
x ;;! Hi (x0 M ) ;! Hi (x M ): n
These modules are nite. If Hi (x M ) = 0, then multiplication by xn on Hi (x0 M ) is surjective, whence Hi (x0 M ) = 0 by Nakayama's lemma. Corollary 1.6.19. Let (R ) be a Noetherian local ring, M 6= 0 a nite Rmodule, and I an ideal generated by x = x1 . . . xn. Then the following m
m
are equivalent: (a) grade(I M ) = n (b) Hi (x M ) = 0 for i > 0 (c) H1 (x M ) = 0 (d) x is an M-sequence.
The equivalence of (a) and (b) follows from 1.6.17, and (b) ) (c) and (d) ) (a) are trivial. The proof of (c) ) (d) is an easy induction based on 1.6.18 and 1.6.13. We saw in 1.1.6 that under the hypotheses of 1.6.19 every permutation of an M -sequence is again an M -sequence. Since, by 1.6.8, the Koszul complexes of x and every permutation of x are isomorphic, 1.6.19 yields another proof of 1.1.6. Remark 1.6.20. For an arbitrary ring R and an arbitrary module M it follows from H1 (x M ) = 0 that x is M -quasi-regular, provided xM 6= M see 48], Ch. X, x9, Theor!eme 1. Proof.
The Koszul complex as an invariant. Let R be a Noetherian local ring, I an ideal, and x = x1 . . . xn and y = y1 . . . yn minimal systems of generators of I . Then any n n matrix A = (apq) such that
xi =
Xn j =1
ajiyj
i = 1 . . . n
is invertible since the residue classes of x and y are bases of I= I over R= . If f and f 0 are the linear forms on R n de ned by x and y respectively, then there exists an R -automorphism ' of R n (de ned by A) such that f = f 0 ', and it follows from 1.6.8 that the Koszul complexes K (x) and K (y ) are isomorphic. This obviously fails if x and y have m
m
.
.
53
1.6. The Koszul complex
di erent lengths. Nevertheless the Koszul complexes K (x) and K (y ) are closely related. The following proposition shows how to compare each of them to K (x y ). Proposition 1.6.21. Let R be a ring, x = x1 . . . xn a sequence in R, and x0 = x1 . . . xn xn+1 . . . xm with xn+1 . . . xm 2 (x). Then .
.
.
K. (x0 ) = K. (x)
V
^ Rm;n
as graded R-algebras here R m;n is considered a complex with zero differential. In particular, for every R-module M one has
H. (x0 M ) = H. (x M )
V
V
^ Rm;n:
V
Since PR k+1 = R k R it suces to treat the case m = n + 1. n Write xn+1 = j =1 aj xj . Let f be the linear form on R n+1 de ned by x0 and f 0 the linear form de ned by x00 = x 0. The assignment ei 7! ei for P n i = 1 . . . n and en+1 7! j =1 aj ej + en+1 induces an automorphism ' of R n+1 such that f = f 0 '. As above one concludes that K (x0 ) = K (x00) in other words, there is no restriction in assuming that xn+1 = 0. In the special situation we have reached, the rst claim is a trivial consequence of 1.6.6. The second claim is easily veri ed. Corollary 1.6.22. Let R be a ring, I a nitely generated ideal, and M an Rmodule. Suppose x = x1 . . . xm and y = y1 . . . yn are systems of generators of I, and let g 2 N. Then Hi (x M ) = 0 for i = m ; g + 1 . . . m if and only if Hj (y M ) = 0 for j = n ; g + 1 . . . n. The corollary follows easily from 1.6.21. Note that for a nite module M over a Noetherian ring R it just restates part of 1.6.17. However, when we de ne the grade of a nitely generated ideal with respect to an arbitrary module in Chapter 9, 1.6.22 will be an essential result. Proof.
.
Exercises
.
f
g
Let I , J , I1 , I2 , I3 be nite subsets of N. Suppose I = i1 . . . ip , J = ip+1 . . . ip+q , the elements given in ascending order. (a) Suppose I J = , and let be the permutation of I J given by (jk ) = ik where I J = j1 . . . jp+q is given in ascending order. Prove (I J ) = () = ( 1)pq (J I ). (b) Deduce that (I1 I2 ) (I1 I2 I3 ) = (I1 I2 I3 ) (I2 I3 ). 1.6.24. Let R be a local ring, and M a nite R -module. ; V (a) Show ( i M ) = (Mi ) for all i N. V (b) Let 1 i (M ). Prove that M is free if and only if i M is free. 1.6.23.
f
;
g
\ f g
2
54
1. Regular sequences and depth ;
V
(a) Let R be a ring, and M an R -module of rank r. Prove rank i M = ri for all i 2 N. (b) Show the analogue for a homomorphism ' : F ! G of nite free modules over a Noetherian ring. Hint for (b): One may assume that R is local and of depth 0. Then Im ' is a free direct summand of G. 1.6.26. Let R be a Noetherian local ring, F a nite free R -module, U F a V submodule ofV rank r and the natural embedding. Show that j is V injective if V and only if j U is torsion-free. In particular j U is non-zero, but j is not injective for rank U < j (U ). 1.6.27. Let R be a ring, and M an RV -module. For f1 . . . fp 2 M let '(f1 . . . fp ) p be the restriction of d . . . d to M . Show that ' induces an R -linear map f f p Vp Vp : (M ) ! ( M ) . Prove that is an isomorphism, if M is nite and free. 1.6.28. Let L be an R -module, x 2 L , and the right multiplication by x on V L . Prove 1.6.25.
1
0
0
0
e . (x): 0 K
2
^ ;;! R ;;! L ;;! L ;;! 0
0
is a complex. Suppose that L = (R n ) and f 2 L . Then the complexes Ke . (f ) and K . (f ) are isomorphic. (Since K . (f )
= K. (f ) by 1.6.10, one can introduce the Koszul complex via Ke . (f ) if one is satis ed with having it only for linear forms on nite free modules.) 1.6.29. Let R be a Noetherian ring, x = (x1 . . . xn ) an element of R n , M = R n =Rx, and I the ideal generated by x1 . . . xn . Prove that grade I k if and only if 0
0
0 ;;! R ;;! R n ;;!
2 ^
Rn
V
k
^ n ;;! ;;! R ;;! 0
is a free resolution of k M . (The map
is right multiplication by x as in 1.6.28 Vk V V one always has M
= ( k R n )= ( k 1 R n ).) 1.6.30. Let x = x1 . . . xn be a sequence in R , and denote by 'i the di erential ; Vi n V R ! i 1 R n in the Koszul complex of x. Let ri = ni 11 be the expected rank of 'i . (a) Show that Rad Iri ('i ) = Rad(x). (b) Derive 1.6.17 for M = R from the Buchsbaum{Eisenbud acyclicity criterion. 1.6.31. Let R be a Noetherian ring, and M 6= 0 a nite R -module. Let I be an ideal, x = x1 . . . xn a system of generators of I , and g = grade(I M ). Show Hi (x M ) = 0 for i = n ; g + 1 . . . n, and Hi (x M ) 6= 0 for i = 0 . . . n ; g. (This property is called the rigidity of the Koszul complex.) Hint: Reduce to the local case and use 1.6.18 for an inductive argument. 1.6.32. Let (R ) be a Noetherian local ring, I an ideal, x 2 , and M a nite R -module. Prove grade(I + (x) M ) grade(I M ) + 1. 1.6.33. Let (R ) be a local ring, and 6= be a prime ideal such that = . Choose a 2 with = + (a) (see the proof of 1.5.9). Then a is R -regular (why?). If F. is a graded minimal free resolution of R= , show that (F. K. (a)) is a minimal free resolution of R = R . ;
;
;
;
m
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Notes
55
Notes After the foundations of homological algebra had been laid by Cartan and Eilenberg 67], it invaded commutative ring theory through the epochal work of Auslander and Buchsbaum 17], 18], 19], Rees 303], and Serre 332]. These works cover the contents of Sections 1.1{1.3, and much more, to be developed in Chapters 2{4. Previously commutative algebra had been ideal theory (under which title Krull (in German) and Northcott published inuential monographs) now modules were considered the objects that give structure to a ring. An intermediate position was taken by Grobner's rather `modern' treatise 141], but it introduced modules only as `Vektormoduln', i.e. submodules of free modules over polynomial rings. Proposition 1.2.16 and several theorems in Chapters 2 and 3 resemble a very successful method in topology, namely to relate the properties of the total space of a bration to those of the base and the bre. The algebraic analogue of this principle was studied systematically by Grothendieck 142] (which, by the way, contains various results on regular sequences not reproduced by us). Torsion-freeness, reexivity, and their `higher' analogues are treated in the monograph 16] of Auslander and Bridger see Bruns and Vetter 61] for a compact presentation. The de nition of rank is taken from Scheja and Storch 323]. The very useful acyclicity criterion of Buchsbaum and Eisenbud appeared in 63]. It is closely related to Peskine and Szpiro's equally important `lemme d'acyclicite' 297] which we reproduced in Exercise 1.4.24. The notion of perfect ideal or module appeared in Rees 303]. It is an abstract version of Grobner's 141] which in turn goes back to Macaulay 262]. A special form of the Hilbert{Burch theorem was proved by Hilbert 171] (and had been previously conjectured by Meyer 274]) whereas Burch 66] provided the rst `modern' version. The theorem has been re-proved several times we have essentially reproduced the version of Buchsbaum and Eisenbud 64] who generalized the theorem to a factorization theorem for the ideals Iri ('i) appearing in their acyclicity criterion. Because of their importance for algebraic geometry, graded rings have been a standard topic in commutative algebra. Their enumerative theory will be developed in Chapter 4. Rees 305] ascribes Theorem 1.5.5 to Samuel. Theorem 1.5.8 is due to Matijevic 267], and 1.5.9 was given by Matijevic and Roberts 268]. (The proof of 1.5.9 has been drawn from Fossum and Foxby 109] and Goto and Watanabe 134].) These theorems are part of a programme aiming at characterizations of graded rings which only use localizations with respect to graded prime ideals.
56
1. Regular sequences and depth
We shall reproduce the pertinent results in the exercises of Chapters 2 and 3. The Koszul complex 240] appeared for the rst time in Hilbert 171]: after having proved his syzygy theorem (see 2.2.14) Hilbert determined the free resolution of the kX1 . . . Xn]-module k. That the Koszul complex is an utterly useful construction even when it is not acyclic seems to have been recognized by Auslander and Buchsbaum 19] and Serre 334]. Auslander and Buchsbaum established the main results of Section 1.6 whereas Serre found the connection with multiplicity theory see Chapter 4.
2 Cohen{Macaulay rings
In this chapter we introduce the class of Cohen{Macaulay rings and two subclasses, the regular rings and the complete intersections. The de nition of Cohen{Macaulay ring is suciently general to allow a wealth of examples in algebraic geometry, invariant theory, and combinatorics. On the other hand it is suciently strict to admit a rich theory: in the words of Hochster, `life is really worth living' in a Cohen{Macaulay ring (183], p. 887). The notion of Cohen{Macaulay ring is a workhorse of commutative algebra. Regular local rings are abstract versions of polynomial or power series rings over a eld. The fascination of their theory stems from a unique interplay of homological algebra and arithmetic. Complete intersections arise as residue class rings of regular rings modulo regular sequences, and, in a sense, are the best singular rings. Their exploration is dominated by methods related to the Koszul complex. 2.1 Cohen{Macaulay rings and modules Let R be a Noetherian local ring, and M a nite module. If the `algebraic' invariant depth M equals the `geometric' invariant dim M , then M is called a Cohen{Macaulay module: De nition 2.1.1. Let R be a Noetherian local ring. A nite R -module M 6= 0 is a Cohen{Macaulay module if depth M = dim M . If R itself is a Cohen{Macaulay module, then it is called a Cohen{Macaulay ring. A maximal Cohen{Macaulay module is a Cohen{Macaulay module M such that dim M = dim R . In general, if R is an arbitrary Noetherian ring, then M is a Cohen{ Macaulay module if M is a Cohen{Macaulay module for all maximal ideals 2 Supp M . (So we consider the zero module to be Cohen{ Macaulay.) However, for M to be a maximal Cohen{Macaulay module we require that M is such an R -module for each maximal ideal of R . As in the local case, R is a Cohen{Macaulay ring if it is a Cohen{ Macaulay module. If I is an ideal contained in Ann M , then it is irrelevant for the Cohen{Macaulay property whether we consider M as an R -module or m
m
m
m
m
57
58
2. Cohen{Macaulay rings
an R=I -module. In particular, if R is local and M a Cohen{Macaulay module, then M is a maximal Cohen{Macaulay module over R= Ann M . The next theorem exhibits the fact that for a Cohen{Macaulay module the grade of an arbitrary ideal is given by its `codimension'. Theorem 2.1.2. Let (R ) be a Noetherian local ring, and M 6= 0 a m
Cohen{Macaulay R-module. Then (a) dim R= = depth M for all Ass M, (b) grade(I M ) = dim M dim M=IM for all ideals I , (c) x = x1 . . . xr is an M-sequence if and only if dim M=xM = dim M r, (d) x is an M-sequence if and only if it is part of a system of parameters of M. Proof. (a) We saw depth M dim R= in 1.2.13, and dim R= dim M holds since Ass M Supp M . (b) If grade(I M ) = 0, then there exists Ass M with I therefore dim M=IM = dim M follows from (a). If grade(I M ) > 0, then we choose x I regular on M . One has grade(I M=xM ) = grade(I M ) 1, depth M=xM = depth M 1, and dim M=xM = dim M 1
;
p
p
2
p
p
2
p
2
;
;
m
;
p
;
so that induction completes the argument. (c) It suces now to quote 1.6.19. (d) This is just a reformulation of (c). The Cohen{Macaulay property is stable under specialization and localization: Theorem 2.1.3. Let R be a Noetherian ring, and M a nite R-module. (a) Suppose x is an M-sequence. If M is a Cohen{Macaulay module, then M=xM is Cohen{Macaulay (over R or R=(x)). The converse holds if R is
local. (b) Suppose that M is Cohen{Macaulay. Then for every multiplicatively closed set S in R the localized module MS is also Cohen{Macaulay. In particular, M is Cohen{Macaulay for every Spec R. If M = 0, then depth M = grade( M ) if in addition R is local, then dim M = dim M + dim M= M.
2
p
p
p
6
p
p
p
p
(a) By the de nition of Cohen{Macaulay module one may evidently assume that R is local. Let n be the length of x. Then dim M=xM = dim M ; n by 1.2.12 and depth M=xM = depth M ; n by 1.2.10. (b) Let be a maximal ideal of RS . The ideal is the extension of a prime ideal in R , and so (RS ) = R . Let be a maximal ideal of R containing . Then R is a localization of the Cohen{Macaulay local ring R . So we may again assume that R is local. There is nothing to prove if M = 0. When M 6= 0, we use induction on depth M . If depth M = 0, then 2 Ass M , and is a minimal Proof.
q
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59
2.1. Cohen{Macaulay rings and modules
prime of Supp M by 2.1.2 therefore dim M = 0. The same argument shows that cannot be contained in any 2 Ass M if depth M > 0. So contains an M -regular element x, and the induction hypothesis applies to M=xM . It follows easily that M is Cohen{Macaulay and that depth M = grade( M ). The second equation results from that and 2.1.2. Corollary 2.1.4. Let R be a Cohen{Macaulay ring, and I 6= R an ideal. Then grade I = height I, and if R is local, height I + dim R=I = dim R. Proof. One has height I = minfdim R : 2 V (I )g and furthermore grade I = minfdepth R : 2 V (I )g. Theorem 2.1.3 yields dim R = depth R for all 2 Spec R . This proves the rst equation, and the second follows from that and 2.1.2. Let k be a eld. We shall see in the next section that every nite module over a polynomial ring kX1 . . . Xn] or a power series ring kX1 . . . Xn]] has nite projective dimension. Furthermore these rings are Cohen{Macaulay as will be shown below. This explains why the following theorem is a very e ective Cohen{Macaulay criterion. Theorem 2.1.5. Let R be a Cohen{Macaulay ring, and M a nite R-module p
p
q
p
p
p
p
p
p
p
p
p
p
p
p
of nite projective dimension. (a) If M is perfect, then it is a Cohen{Macaulay module. (b) The converse holds when R is local. Proof. Let M be perfect and Supp M . Then M is a perfect module as shown in the proof of 1.4.16. So we may assume that R is local. The Auslander{Buchsbaum formula gives proj dim M = dim R depth M , and 2.1.4 yields grade M = dim R dim M . Thus depth M = dim M if and only if proj dim M = grade M . One says that an ideal I is unmixed if I has no embedded prime divisors or, in modern language, if the associated prime ideals of R=I are the minimal prime ideals of I . Macaulay showed in 1916 that an ideal I = (x1 . . . xn ) of height n in a polynomial ring over a eld is unmixed, and for regular local rings this was proved by Cohen in 1946. (An ngenerated ideal of height n is said to be of the principal class.) These facts
2
p
p
;
;
and the following theorem explain the nomenclature `Cohen{Macaulay'. Theorem 2.1.6. A Noetherian ring R is Cohen{Macaulay if and only if every ideal I generated by height I elements is unmixed. Proof. `)': Suppose I = (x), x = x1 . . . xn , and let 2 Ass R=I , . Then there is a maximal ideal with , and R , R 2 Ass(R =I ). If height I = n, then dim(R =I ) = dim R ; n, and x is an R -sequence by 2.1.2. Therefore R =I is Cohen{Macaulay, whence R = R (again by 2.1.2), and so = . p
m
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60
2. Cohen{Macaulay rings
`(': Let J R be an arbitrary ideal, say height J = n. Then there exist x1 . . . xn 2 J with height(x1 . . . xi ) = i for all i = 0 . . . n (see A.2). It is impossible for xi+1 to be contained in a minimal prime ideal of (x1 . . . xi ). By hypothesis it therefore is an (R=(x1 . . . xi ))regular element. So x1 . . . xn is an R -sequence. We have shown that grade J = height J for every proper ideal J of R . Then R is certainly Cohen{Macaulay. Flat extensions of Cohen{Macaulay rings and modules. The behaviour of
depth under at local extensions was studied in Section 1.2. That makes it easy to prove an analogous theorem for the Cohen{Macaulay property. Theorem 2.1.7. Let ' : (R ) ! (S ) be a homomorphism of Noetherian m
n
local rings. Suppose M is a nite R-module and N is an R-at nite Smodule. Then M R N is a Cohen{Macaulay S-module if and only if M is Cohen{Macaulay over R and N= N is Cohen{Macaulay over S.
m
In fact, according to 1.2.16 we have depthS M N = depthR M + depthS N= N . Since depth is bounded above by dimension, the theorem follows from the analogous equation for dimension see A.11. Corollary 2.1.8. Let (R ) be a Noetherian local ring, M a nite R^ its -adic completion. module, and M ^ (a) Then dimR M = dimR^ M^ and depthR M = depthR^ M. (b) M is Cohen{Macaulay if and only if M^ is Cohen{Macaulay. ^ is local and at, and M^ = M R R^ since M Proof. The extension R ! R is nite. One can of course use more direct arguments in order to prove the previous corollary. Similarly there is a more `elementary' approach to the following theorem see for example 231]. Theorem 2.1.9. Let R be a Noetherian ring, M a nite R-module, and S = R X1 . . . Xn ] or S = R X1 . . . Xn]]. Then M S is a Cohen{Macaulay m
m
m
S-module if and only if M is a Cohen{Macaulay module.
Since the indeterminates can be adjoined successively, we may assume n = 1, X = X1. The `only if' part is easy: in both cases X is (M S )-regular, and R = S=(X ), M = (M S )=X (M S ). (That X is (M S )-regular is evident for S = R X ] the reader should nd a justi cation for S = R X ]].) Conversely, let be a maximal ideal of S , and set = \ R . As outlined below A.12 the bre S = S is a discrete valuation ring, and thus Cohen{Macaulay. Now we invoke 2.1.7 and complete the proof. Proof.
m
p
p
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m
61
2.1. Cohen{Macaulay rings and modules
For polynomial extensions the proof of 2.1.9 shows that a stronger local version of 2.1.9 is valid: for 2 Spec R X1 . . . Xn ] the localization R X1 . . . Xn ] is Cohen{Macaulay if and only if R is Cohen{Macaulay for = \ R . Similarly, there is a local version of the following theorem: Theorem 2.1.10. Let k be a eld, R a Noetherian k-algebra, and K an q
q
p
p
q
extension eld of k. Suppose that R is a nitely generated k-algebra, or that K is nitely generated as an extension eld of k. Then R is a Cohen{ Macaulay ring if and only if R k K is. Proof. If R is a nitely generated k-algebra, then R k K is a nitely generated K -algebra, and therefore Noetherian. Suppose that K is a nitely generated extension eld. Then K is a nite algebraic extension of a nite purely transcendental extension K 0 of k. Since K 0 is the eld of fractions of a polynomial ring kT1 . . . Tn], we nd again that R k K 0 is Noetherian, whence R k K = (R k K 0 ) K 0 K is also Noetherian. Evidently R K is a faithfully at R -algebra. Therefore, given a prime ideal of R , there exists Spec R K such that = R . The bre of the extension R (R K ) is a localization of k( ) K . In conjunction with 2.1.7 this argument reduces the theorem to the assertion that L k K is Cohen{Macaulay for extension elds L and K of k, provided one of
!
p
q
2
p
p
p
q
\
q
them is nitely generated. This follows from the next proposition. Proposition 2.1.11. Let k be a eld, R a k-algebra, and K a nitely generated extension eld of k. Then R
k K is isomorphic to a ring
R X1 . . . Xn ]S =(f1 . . . fm ) where S is a multiplicatively closed subset of R X1 . . . Xn], and f1 . . . fm is a R X1 . . . Xn]S -sequence. Proof. The extension k K decomposes into a series of cyclic extensions k = K0 Kt = K . We use induction on t. Suppose that T = R k Ki = R X1 . . . Xn]S =(f1 . . . fm ). If Ki+1 = Ki Y ]=(g) with a monic irreducible polynomial g, then
R
k Ki+1 = T K Ki+1 = T Y ]=(g): i
Since T is a at Ki -algebra, g is not a zero-divisor of T Y ] = R X1 . . . Xm Y ]S =(f1 . . . fm ): If Ki+1 = Ki (Y ), then R k Ki+1 = R X1 . . . Xn Y ]S 0 =(f1 . . . fm ) where S 0 is generated by the image of S and the image of Ki Y ] n f0g. Chain conditions in Cohen{Macaulay rings. Cohen{Macaulay rings were
introduced as those rings for which depth equals dimension. Corollary 2.1.4 and the next theorem show that dimension theory itself is simpler
62
2. Cohen{Macaulay rings
for Cohen{Macaulay rings than for general Noetherian rings. One says a Noetherian ring R is catenary if every saturated chain joining prime ideals and , , has (maximal) length height = R is universally catenary if all the polynomial rings R X1 . . . Xn ] are catenary. It is easy to see that R is universally catenary if and only if every nitely generated R -algebra is (universally) catenary. Theorem 2.1.12. A Cohen{Macaulay ring R is universally catenary. Proof. `Universally' may be dropped because of 2.1.9. So let be prime ideals of R . The localization R is Cohen{Macaulay, and 2.1.4 applied to R yields height = dim R = height R + dim(R = R ) = height + height = : It is an easy exercise to show that R is catenary if this equation holds for all prime ideals . Corollary 2.1.13. A Noetherian complete local ring R is universally catep
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nary.
Cohen's structure theorem (see A.21) tells us that R is a residue class ring of a formal power series ring A = kX1 . . . Xn]] where k is a eld or a discrete valuation ring. By 2.1.9, A is Cohen{Macaulay and therefore universally catenary R inherits this property as a residue class ring of A. Remark 2.1.14. For the sake of clarity 2.1.4 and 2.1.12 were kept more special than necessary. If R has a Cohen{Macaulay module M with Supp M = Spec R , then we need only replace grade I by grade(I M ) in 2.1.4 to obtain an equally valid result. It follows that a local Noetherian domain which has a maximal Cohen{Macaulay module is universally catenary. One of Nagata's famous counter-examples is a non-catenary such domain (284], Example 2, p. 203). However, to be universally catenary is not the only necessary condition for R to have a Cohen{Macaulay module M with Supp M = Spec R it must also satisfy Grothendieck's condition (CMU). This condition requires that for every prime ideal of R the spectrum of R= contains a non-empty open subset U such that (R= ) is Cohen{Macaulay for all 2 U see 142], IV, 6.11. A local ring violating (CMU) was constructed by Ferrand and Raynaud 106]. A Noetherian complete local ring is universally catenary since it is a residue class ring of a Cohen{Macaulay ring, and for the same reason it satis es (CMU). It is an open question whether every Noetherian complete local ring has a maximal Cohen{Macaulay module. We shall see in Chapter 9 that the existence of maximal Cohen{ Macaulay modules implies a wealth of homological theorems. Fortunately, it will not be essential that these Cohen{Macaulay modules M Proof.
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2.1. Cohen{Macaulay rings and modules
are really nite we `only' need every system of parameters of the ring to be an M -sequence. In Chapter 8 such modules will be shown to exist for local rings containing a eld. For a prime ideal in a Cohen{Macaulay local ring R the residue class ring R= is not Cohen{Macaulay in general it is however unmixed in the sense of Nagata 284]: Theorem 2.1.15. Let R be a Cohen{Macaulay local ring, and a prime ^ = dim R= for all 2 Ass(R= ^ R^ ). In particular R^ ideal. Then dim R= p
p
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p
is an unmixed ideal. ^ R^ )-regular element. Proof. If R = , then would contain an (R= Therefore R = , and we have a at local ring extension R R^ . Applying 1.2.16 and since R^ and R are Cohen{Macaulay we get
\ 6 \
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dim R^ = depth R^ = depth R + depth(R^ = R^ ) = dim R : In view of 2.1.4 this equation is equivalent to the theorem. p
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Serre's condition (Sn).
Sometimes one only needs a ring or a module to be Cohen{Macaulay in low `codimension'. A nite module over a Noetherian ring R satis es Serre's condition (Sn) if depth M min(n dim M ) for all 2 Spec R . The theorems of this section need some modi cation when Cohen{Macaulay is replaced by (Sn ). As an example we treat the (Sn) analogue of 2.1.7. Proposition 2.1.16. Let ' : R ! S be a at homomorphism of Noetherian p
p
p
rings. (a) Let Spec S and = R. If S satis es (Sn), then so does R . (b) Suppose R and all the bres k( ) S with Spec R satisfy (Sn). Then S satis es (Sn ). Proof. (a) Replacing R by R and S by S we may assume that ' is a at homomorphism of local rings. For Spec R we now choose a minimal prime of S . Then dim(S = S ) = 0, and according to 1.2.16 we have q
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depth R = depth S min(n dim S ) = min(n dim R ): (b) For 2 Spec S and = R \ one similarly deduces depth S = depth R + depth(S = S ) min(n dim R ) + min(n dim(S = S )) min(n dim R + dim(S = S )) = min(n dim S ): q
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2. Cohen{Macaulay rings
Exercises Let k be a eld, and R = k X1 . . . Xn ]. If is a prime ideal in R with 2 f0 1 n ; 1 ng, show that R= is Cohen{Macaulay. 2.1.18. Let k be a eld. Show (a) the subalgebra S = k U 4 U 3 V UV 3 V 4 ] of k U V ] is not Cohen{Macaulay, (b) for each m with 2 m n ; 2 there exists a prime ideal of height m in R = k X1 . . . Xn ] for which R= is not Cohen{Macaulay. 2.1.19. Let k be a eld, and S the subalgebra of k X1 . . . Xn ] generated by the monomials of degrees 2 and 3. Show S is an n-dimensional domain the maximal ideal (X1 . . . Xn ) \ S has height n and grade 1. 2.1.20. Prove (a) a one dimensional reduced Noetherian ring is Cohen{Macaulay, (b) a one dimensional Noetherian local ring has a maximal Cohen{Macaulay module. 2.1.21. Characterize (Sn ) by an unmixedness property. 2.1.22. Prove that a module M satis es (Sn ) if and only if M is Cohen{Macaulay for all prime ideals with depth M < n. 2.1.23. Let R ! S be a faithfully at homomorphism of Noetherian rings. Show the following are equivalent: (a) S is Cohen{Macaulay (b) R and all the bres S = S are Cohen{Macaulay where 2 Spec S and = \ R. Hint: use A.10. 2.1.24. Prove the analogues of 2.1.8(b), 2.1.9, and 2.1.10 for (Sn ). For the passages from R to R^ and to R
X1 . . . Xn ]] assume that R is a residue class ring of a Cohen{Macaulay ring. 2.1.25. Prove the converse of 2.1.5(a) under the hypothesis that Supp M is connected. (The crucial point is to show that the function 7! proj dim M is locally constant on Supp M if M is locally perfect.) 2.1.26. Let R be a Cohen{Macaulay local ring of dimension d and M a nite R -module. Deduce that the d -th syzygy of M in an arbitrary nite free resolution is either 0 or a maximal Cohen{Macaulay module. 2.1.27. Let R be a Noetherian graded ring, and M a nite graded R -module. Show: (a) For 2 Spec R the localization M is Cohen{Macaulay if and only if M is. (This follows easily from the results of Section 1.5.) (b) The following are equivalent: (i) M is Cohen{Macaulay (ii) M is Cohen{Macaulay for all graded prime ideals (iii) M( ) is Cohen{Macaulay for all graded prime ideals . (c) Suppose in addition that (R ) is local. Then M is Cohen{Macaulay if and only if M is. 2.1.28. Let (R ) be a Noetherian local ring, and x 2 a homogeneous R -regular element. Then R is Cohen{Macaulay if and only if so is R=(x). 2.1.17.
height
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65
2.2. Regular rings and normal rings
Let the Noetherian ring R be a free Z-module such that R K is Cohen{Macaulay for some eld K of characteristic p > 0. Show that R L is Cohen{Macaulay for every eld L of characteristic 0. Hint: reduce the problem to the case in which K = Z=(p), L = Q and use R Z , = (p), as a `bridge'. This is the rst and easiest example of reduction to characteristic p.
2.1.29.
p
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2.2 Regular rings and normal rings The most distinguished of all Noetherian local rings are those whose maximal ideal can be generated by a system of parameters: De nition 2.2.1. A Noetherian local ring (R ) is regular if it has a system of parameters generating such a system of parameters is called a regular system of parameters. Evidently, when dim R = 0, then R is regular if and only if it is a eld, and when dim R = 1, R is regular if and only if it is a discrete valuation ring. Other examples of regular local rings are kX1 . . . Xn ]] where k is a eld, and kX1 . . . Xn] , = (X1 . . . Xn ). We may rephrase the de nition above as follows: R is regular if and only if ( ) = dim R . In fact, ( ) dim R by Krull's principal ideal theorem, and a system of generators of has dim R elements exactly when it is a system of parameters. Proposition 2.2.2. A Noetherian local ring (R ) is regular if and only if its -adic completion R^ is regular. Proof. The maximal ideal of R^ is R^ , and we have natural isomor^ R^ , = 2 phisms R= = R= = ( R^ )=( R^ )2. Therefore ( ) = ( R^ ). ^ Furthermore dim R = dim R , and by de nition R is regular if and only if dim R = ( ). It is easily proved that regular local rings are integral domains. Proposition 2.2.3. Let (R ) be a regular local ring. Then R is an integral m
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domain.
We use induction on dim R . When dim R = 0, R is a eld. So suppose dim R > 0, and let 1 . . . m be the minimal prime ideals of R . There exists an element x 2 which is not contained in any of the ideals 2 . . . . (This follows easily from 1.2.2 with M = N = .) Since x is 1 m part of a minimal system of generators of , it is part of a regular system of parameters, and thus R=(x) is regular (use that dim R=(x) = dim R ; 1). As dim R=(x) < dim R we may assume that R=(x) is a domain. Thus (x) is a prime ideal, and therefore contains a minimal prime ideal of R , say = 1 , z is an element 1 . Every y 2 1 has the form y = xz , and since x 2 of 1. It follows that 1 = x 1, which, by Nakayama's lemma, implies 1 = 0, as required.
Proof.
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2. Cohen{Macaulay rings
Using the previous proposition, one can say precisely which residue class rings of a regular local ring are also regular: Proposition 2.2.4. Let R be a regular local ring, and I R an ideal. Then
R=I is regular if and only if I is generated by a subset of a regular system of parameters. Proof. The `if' part is trivial. So suppose that R=I is regular. Then ( =I ) = dim R=I set m = dim R dim R=I . By Nakayama's lemma I contains elements x1 . . . xm which are part of a minimal system of generators of . Then R=(x1 . . . xm ) is regular of dimension dim R m = dim R=I . Since I and (x1 . . . xm ) are prime ideals, one must have I = (x1 . . . xm).
;
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;
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The next proposition gives useful characterizations of regularity. Proposition 2.2.5. Let (R k) be a Noetherian local ring, and x1 . . . xn a m
minimal system of generators of . Then the following are equivalent: (a) R is regular (b) x1 . . . xn is an R-sequence (c) the substitution Xi x i = 2 yields an isomorphism kX1 . . . Xn] = gr (R ). Proof. (a) (b): Since x1 . . . xn is a minimal system of generators of , it is a regular system of parameters, and R=(x1 . . . xi) is also regular for each i. Therefore R=(x1 . . . xi ) is a domain, and xi+1 is regular on R=(x1 . . . xi ). (b) (a): An R -sequence is part of a system of parameters by 1.2.12. m
7! 2
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)
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)
(b) () (c): This follows from 1.1.8 and its converse 1.1.15. Corollary 2.2.6. A regular local ring is Cohen{Macaulay.
The Auslander{Buchsbaum{Serre theorem. Whereas the characterizations
of regular local rings in 2.2.5 are rather close to the de nition, this can hardly be said of the following theorem. Together with 2.2.19 below, it is considered to be the most important achievement of the use of homological algebra in the theory of commutative rings. Theorem 2.2.7 (Auslander{Buchsbaum{Serre). Let (R k) be a Noem
therian local ring. Then the following are equivalent: (a) R is regular (b) proj dim M < for every nite R-module M (c) proj dim k < . Proof. (a) (b): Let d = dim R , and N a d -th syzygy module of M . Since R is Cohen{Macaulay, N is a maximal Cohen{Macaulay module or 0 by Exercise 2.1.26. If N = 0, we are done otherwise every regular system of parameters x is a (maximal) N -sequence. Lemma 1.3.5 gives
)
1 1
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2.2. Regular rings and normal rings
proj dimR N = proj dimR=(x) (N=xN ) = proj dimk (N= N ) = 0. So N is free, and proj dim M d . (b) ) (c): This is trivial. (c) ) (a): This is a special case of the following theorem. Theorem 2.2.8 (Ferrand, Vasconcelos). Let (R ) be a Noetherian local ring, and I 6= 0 a proper ideal with proj dim I < 1. If I=I 2 is a free m
m
R=I-module, then I is generated by a regular sequence. Proof. Since I has a nite free resolution, it contains an R -regular element x by 1.4.7. It is no restriction to assume x = I : if x I , then we choose some y I I Ry + Rx is not contained in any Ass R , and by 1.2.2 there is a R for which y + ax has the same property. This proves the theorem when (I ) = 1, and when (I ) > 1, we use induction, passing from R to R=(x) and from I to I=(x). Of course, we must rst verify that proj dimR=(x) I=(x) < . Since x = I , the residue class of x in I=I 2 is part of a basis x x 2 . . . x m of this free module. Set J = (x2 . . . xm ) we claim: J (x) xI . In fact, if z = ax = a2 x2 + + amxm , then a I because x x 2 . . . x m are linearly independent modulo I . Therefore we get a composition of maps
2 n
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\ 2 I=(x) = (J + (x))=(x) = J=J \ (x) ;! I=xI ;! I=(x)
m
in which the residue class of xi is sent to itself, and which therefore is the identity on I=(x). So I=(x) is a direct summand of I=xI as x is I -regular, the latter has nite projective dimension over R=(x) by 1.3.5. Finally we need that I =I 2 is a free R=I -module where I = I=(x). But this is a very easy consequence of the linear independence of x x2 . . . xm modulo I . The proof of 2.2.7 can be varied: the Koszul complex of a regular system of parameters resolves k by 2.2.5 and 1.6.14, whence (a) ) (c). Moreover, the implication (c) ) (b) follows from 1.3.2: proj dim k < 1 ) TorRi (M k) = 0 for i 0, and this in turn gives proj dim M < 1. While this reasoning uses a truly homological argument, namely the fact that Tor can be computed from a free resolution of either module, the proof above merely exploits the existence of minimal free resolutions. Serre's original argument for (c) ) (a) will be indicated in Exercise 2.3.22. Corollary 2.2.9. Let R be a regular local ring, and a prime ideal in R. p
Then R is regular.
By 2.2.7(a) ) (b) we have proj dim R= < 1. It follows that proj dimR (R= ) = proj dimR (R = R ) < 1, whence R is regular by 2.2.7(c) ) (a). Over a regular local ring the Cohen{Macaulay property is equivalent to perfection (see Section 1.4 for this notion): p
Proof.
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2. Cohen{Macaulay rings
Corollary 2.2.10. A nite module M over a regular local ring is Cohen{ Macaulay if and only if it is perfect.
The corollary is an immediate consequence of 2.1.5 and 2.2.7. Let (R ) be a Noetherian local ring. By 2.1.8, R is Cohen{Macaulay if and only if its -adic completion R^ is Cohen{Macaulay. Furthermore, if R contains a eld or is a domain, then it is a nite module over a regular local subring (see A.22). Thus the following proposition may almost be considered a new description of the Cohen{Macaulay property for rings. Proposition 2.2.11. Let R be a Noetherian local ring and S a regular local m
m
subring such that R is a nite S-module. Then R is Cohen{Macaulay if and only if it is a free S-module.
By 2.2.7 one has proj dimS R < 1 therefore R is S -free if and only if depthS R = dim S . Choose a (regular) system of parameters x in S . Then x is also a system of parameters of R , and therefore R is Cohen{Macaulay () x is an R -sequence () depthS R = dim S . (One could also use Exercise 1.2.26.) Proof.
Flat extensions of regular rings. The behaviour of regularity under at
local extensions is described by the following theorem. Theorem 2.2.12. Let ' : (R k) ! (S l ) be a at homomorphism of m
n
Noetherian local rings. (a) If S is regular, then so is R. (b) If R and S= S are regular, then so is S. Proof. (a) Let F. be a minimal free resolution of the R -module k. Then F. S is a free resolution of k S = S= S because of atness, and even a minimal one since '( ) . Thus proj dimR k = proj dimS S= S < , and R is regular by 2.2.7. (b) Let m = dim R , n = dim S= S , and choose minimal systems of generators x1 . . . xm of and y1 . . . yn of = S . Then '(x1) . . . '(xm ), y1 . . . yn generate , and S must be regular because dim S = dim R + dim S= S see A.11. An easy example shows that S= S need not be regular in the situation of 2.2.12(a): Let k be a eld, and choose S = kX Y ]]=(Y X 2) and R = ky]] S . Then R and S are regular, and S is a free R -module generated by 1 and x, but S=yS = kX ]]=(X 2) is not regular. (The reader m
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;
should imagine the geometry of this example.) In order to formulate a theorem relating the regularity of R and that of R X1 . . . Xn] we must rst agree on calling a Noetherian ring R regular if its localizations R with respect to maximal ideals are regular. m
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2.2. Regular rings and normal rings
Theorem 2.2.13. A Noetherian ring R is regular if and only if R X1 . . . Xn] is regular. The same holds for R and R X1 . . . Xn]]. Proof. We may assume that n = 1 and set X = X1. Suppose that R X ] is regular, and, given a maximal ideal of R , choose = ( X ) R X ]. Then X 2= 2 , equivalently X 2= ( R X ] )2 , and it follows immediately from the de nition of regularity that R is regular. The same argument shows that regularity descends from R X ]] to R . Of course, what has just been shown can also be derived from 2.2.12(a), and the reader is invited to use 2.2.12(b) in proving that regularity ascends from R to R X ] and R X ]]. (Compare the proof of 2.1.9.) In particular, a polynomial ring kX1 . . . Xn] over a eld k is regular. Corollary 2.2.14. Let k be a eld, and R = kX1 . . . Xn]. (a) (Hilbert's syzygy theorem) Every nite graded R-module M has a nite m
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graded free resolution of length n. (b) Moreover, proj dim M n for every nite R-module M. (c) In fact, every nite R-module has a nite free resolution of length n. Proof. (a) Set = (X1 . . . Xn) and consider a minimal graded free resolution F. of M . Such a resolution exists, and furthermore F. R is a minimal free resolution of M see 1.5.15. Now R is a regular local ring. Therefore F. R has length at most n, and the same holds true for F. . (b) Consider an arbitrary maximal ideal of R . Then dim R dim R = n, and R is a regular local ring. Hence proj dimR M n. Taking the supremum over all maximal ideals, we get proj dim M n. (In fact, let N be the n-th syzygy of M in a resolution by nite projective R -modules. Then N is a nite free R -module for all , and therefore N
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is projective.) (c) By the theorem of Quillen and Suslin (318], Theorem 4.59) every nite projective R -module is free. In 2.2.14(a) and 2.2.15 below it is not essential that deg Xi = 1 for all i. One may replace the standard grading of R by any grading which makes R a local ring, for example by a grading such that deg Xi > 0 for all i. Corollary 2.2.15. Let k be a eld, R = kX1 . . . Xn ], = (X1 . . . Xn), n
n
n
m
and M a nite graded R-module. Then the following are equivalent: (a) M is Cohen{Macaulay (a0) M is perfect (b) M is Cohen{Macaulay (b0) M is perfect. m
The implications (a0 ) ) (a) ) (b) ) (b0) follow from 2.1.5 and 2.2.10. The remaining implication (b0 ) ) (a0 ) is an immediate consequence m
Proof.
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2. Cohen{Macaulay rings
of the equations proj dim M = proj dim M and grade M = grade M proved in 1.5.15. Remark 2.2.16. Let R be a Noetherian k-algebra where k is a eld, and K an extension eld of k. If R is nitely generated as a k-algebra or K is a nitely generated extension eld, then R k K is a Noetherian ring as shown in the proof of 2.1.10. Since R k K is a at R -algebra, it follows readily from 2.2.12 that R is regular if R k K is regular. We saw in the proof of 2.1.10 that the bres of the extension R ! R k K are of the form (L k K ) where L is an extension eld of k and 2 Spec(L k K ). If L k K is regular for every extension eld L of k (provided one of K L is nitely generated), then one obtains from 2.2.12 that R k K is regular if R is regular. The elds K satisfying the condition just formulated are the separable extensions of k. We refer the reader to 270], x26 for a discussion of separability, and to 142], IV, 6.7.4.1 for the theorem concerning the regularity of L k K . m
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Factoriality of regular local rings. Our next goal is to show that a regular local ring is a factorial domain (a UFD in other terminology). We need two elementary lemmas whose proofs are left as an exercise for the reader. Lemma 2.2.17. A Noetherian domain R is factorial if and only if every prime ideal of height 1 is principal. Lemma 2.2.18. Let R be a Noetherian domain and a prime element in R. p
Then R is factorial if and only if R is factorial.
Theorem 2.2.19 (Auslander{Buchsbaum{Nagata). A regular local ring R is factorial.
We use induction on dim R . If dim R = 0, then R is a eld, and there is nothing to prove. So suppose dim R > 0, and choose 2 n 2 . Since R=( ) is again a regular local ring, is a prime element. According to the previous lemma, it is enough to show that S = R is factorial. Let be a prime ideal of S with height = 1. Every localization S is a localization of R with respect to a prime ideal 6= , and therefore a regular local ring by 2.2.9. By induction S is factorial. If 6 , then S = S for trivial reasons, and if , then also S = S as follows from 2.2.17 in conjunction with the factoriality of S . This implies that is a projective S -module of rank 1. Of course is of the form S with a prime ideal of R . The R -module has a nite free resolution F , whence = has an augmented resolution Proof.
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P
G. : 0
P
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' ' Gs;1 ;! ;! G1 ;! G0 ;! ;! 0 ;! Gs ;! s
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2.2. Regular rings and normal rings
by nite free S -modules. However, is a projective S -module, and its syzygy modules with respect to G are likewise projective. In particular Im 's;1 Gs = Gs;1. If s > 1, we can modify the tail of G to obtain the free resolution ' G : 0 ;! Im 's;1 Gs ;! Gs;2 Gs ;! ;! G1 ;! G0 ;! ;! 0: Therefore, by induction on the length of G , in fact has a free resolution p
.
.
1
.
p
.
p
0 ;! S n ;! S n+1 ;! ;! 0: The Hilbert{Burch theorem 1.4.17 yields that = aIn(') with some a 2 S , and furthermore that = aS since is projective. So is a principal ideal. A ring is normal if all its localizations are integrally closed domains a Noetherian ring is normal if and only if it is the direct product of nitely many integrally closed domains (see 270] for a detailed discussion of normality). Corollary 2.2.20. A regular local ring is a normal domain. A regular ring '
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p
p
p
is the direct product of regular domains. In fact, every factorial ring R is a normal domain. (One proves this just as for the special case R = Z.) The `classical' proof of the corollary
uses 2.2.5 and the fact that a Noetherian local ring is a normal domain if gr (R ) is a normal domain see 4.5.9. There is even a third proof, as we shall see now. m
Serre's normality criterion. A Noetherian ring R satis es Serre's condition (Rn) if R is a regular local ring for all prime ideals in R with dim R n. (Note the similarity with (Sn ) contrary to (Sn) however, (Rn) says nothing about localizations R with dim R > n.) We leave it as an exercise for the reader to prove that the behaviour of (Rn) under at local extensions is the same as that of (Sn): Proposition 2.2.21. Let ' : R ! S be a at homomorphism of Noetherian p
p
p
p
p
rings. (a) Let Spec S and = R. If S satis es (Rn), then so does R . (b) If R and all bres k( ) S, Spec R, satisfy (Rn), then so does S. It is easy to see that a Noetherian ring R is reduced if and only if it satis es (R0) and (S1). Serre characterized normality in a similar way: Theorem 2.2.22 (Serre). A Noetherian ring R is normal if and only if it satis es (R1) and (S2). q
2
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\
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We refer the reader to Serre 334], IV.4, or 270], x23, for a proof of 2.2.22. The following corollary is an evident consequence of 2.1.16, 2.2.21, and 2.2.22:
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2. Cohen{Macaulay rings
Corollary 2.2.23. Let ' : R
! \
S be a at homomorphism of Noetherian rings. (a) Let Spec S and = R. If S is normal, then so is R . (b) If R and all the bres k( ) S, Spec R, are normal, then so is S.
2
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Suppose that (R ) and (S ) are local, and that ' is at and local. Then, for S to be normal, it is not sucient to have R and S= S normal: there are normal local domains whose completions are not even domains see 284], p. 209, Example 7. m
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Exercises Let R be a Noetherian graded ring. Show: (a) For 2 Spec R the localization R is regular if and only if R is. (b) The following are equivalent: (i) R is regular (ii) R is regular for all graded prime ideals (iii) R( ) is regular for all graded prime ideals . (c) Suppose moreover that (R ) is local. Then R is regular if and only if R is. Hint: Use 1.6.33. 2.2.25. Let R be a positively graded k-algebra over a eld k. Prove the following are equivalent: (a) R is regular (b) R is regular where is the maximal ideal (c) there exist homogeneous elements x1 . . . xn of positive degree for which the assignment Xi 7! xi induces an isomorphism k X1 . . . Xn ]
= R. Hint: For the non-trivial implication (b) ) (c) choose a minimal homogeneous system of generators x1 . . . xn of then apply 1.5.15 and 1.5.4. The rest is a simple dimension argument. 2.2.26. In the situation of 2.2.11 characterize the Cohen{Macaulay R -modules by a property they have as S -modules. 2.2.27. Let R be a Noetherian ring over which every nite module has a nite free resolution. Show R is a factorial domain. 2.2.28. Let R be a regular local ring, and I an ideal of height 1. Prove that the following are equivalent: (a) R=I is Cohen{Macaulay (b) height = 1 for all prime ideals 2 Ass R=I (c) I is a principal ideal. Hint: For (b) ) (c) one uses primary decomposition and the factoriality of R . 2.2.29. Prove that a Noetherian ring R satis es (Ri ) and (Si+1 ) if and only if R is regular for every prime ideal such that depth R i. 2.2.30. (a) Show a Noetherian normal ring of dimension 2 is Cohen{Macaulay. (b) A Cohen{Macaulay ring is normal if and only if it satis es (R1 ). 2.2.31. (a) Let R be a Noetherian complete local domain. Then R is a nite module over a regular local ring S contained in R see A.22. Set M = HomS (R S ) then 2.2.24.
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2.3. Complete intersections M is an R -module in a natural way. Show that depthR M = depthS M min(dim R 2).
(b) Prove that every Noetherian complete local ring of dimension 2 has a maximal Cohen{Macaulay module. 2.2.32. Let R be a Noetherian complete local domain. It is known that the integral closure of R in its eld of fractions is a nite R -module ( 284], (32.1) or 47], Ch. IX, x4). Use this to give a fresh proof of the fact that a Noetherian complete local ring of dimension 2 has a maximal Cohen{Macaulay module. 2.2.33. Let R be a Noetherian ring, and x 2 R an R -regular element. (a) Assume that Rx ful lls (Rn ) and (Sn+1 ), and that (Rn 1 ) and (Sn ) hold for R=(x). Show that R satis es (Rn ) and (Sn+1 ). (b) Assume Rx is a normal domain, and R=(x) is reduced. Show that R is a normal domain. 2.2.34. Let R be a Noetherian graded ring, and A its dehomogenization with respect to an element of degree 1 (see 1.5.18). Show that if R is a normal domain, then so is A. 2.2.35. We keep the notation of 2.2.34. Let be a graded prime ideal of R with x 2= , and 2 Spec A its dehomogenization (see 1.5.26). Show that R is a at local extension of A , and determine its bre. Compare R and S with respect to the following quantities and properties: dimension, depth, type being reduced, an integral domain, Cohen{Macaulay, normal, regular. ;
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2.3 Complete intersections We observed that the homological relationship between a local ring S and a residue class ring R = S=I is particularly strong if I is generated by an S -sequence. In this section we investigate such residue class rings of regular local rings. Slightly more generally we de ne: De nition 2.3.1. A Noetherian local ring R is a complete intersection (ring) if its completion R^ is a residue class ring of a regular local ring S with respect to an ideal generated by an S -sequence. Note that R^ is always a residue class ring of a regular local ring (see A.21). It follows immediately from 2.1.3, 2.1.8, and 2.2.6 that a complete intersection is Cohen{Macaulay. The nomenclature `complete intersection' comes from algebraic geometry. Suppose R is the coordinate ring of an ane variety over an algebraically closed eld k. Then R has the form R = S=I where S is a polynomial ring over k, and R is called a complete intersection if I is generated by the least possible number of elements, namely codim V = height I . Then V is the intersection of codim V hypersurfaces, and I is generated by an S -sequence. Let (S ) be a regular local ring, and (R ) a residue class ring, R = S=I . Suppose that I 6 2 . Then there exists x 2 I , x 2= 2, and we n
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obtain a representation R = S 0 =I 0 with S 0 = S=(x), I 0 = I=(x). The ring S 0 is regular again, and I is generated by a regular sequence if and only if I 0 is: the element x is part of a minimal system of generators of I , and I can be generated by an S -sequence if and only if every minimal system of generators is an S -sequence see 1.6.19. Iterating this procedure we eventually obtain a minimal presentation R = S 00 =I 00 in which S 00 is regular and I 00 ( 00 )2 . It follows that ( ) = ( 00) = dim S 00 . For an arbitrary local ring (R ) the number ( ) is called the embedding dimension of R, emb dim R = ( ): (This terminology is again to be illustrated by the geometric analogue.) The discussion above shows that we may freely assume that I 2 when it is only to be veri ed whether I is generated by an S -sequence or otherwise. Nevertheless our de nition has two aws: rst, it does not use intrinsic characteristics of R second, it is not clear whether for an arbitrary presentation R^ = S=I with S regular local, the ideal I is generated by an S -sequence if R is a complete intersection. The intrinsic characteristics we are seeking are hidden in a Koszul complex. Let us x some standard notation which we shall use frequently throughout this section: (R k) is a Noetherian local ring, and x = x1 . . . xn is a minimal system of generators of . If present, (S k) is a regular local ring such that R = S=I with I 2 the ideal I is minimally generated by a = a1 . . . am , and y = y1 . . . yn is a regular system of parametersP such that xi is the residue class of yi . Furthermore we write ai = ajiyj with aji 2 (necessarily). Let ' : S m ! S n be given by the matrix (aji), and g : S n ! S and h : S m ! S be the linear forms de ned by y and a respectively. Then h = g ', and ^ ' : K (a) ! K (y ) is a complex homomorphism see 1.6.8. By 1.6.7, K (y ) R is just V K (x), and K (a) R has zero di erential forgetting it, we write R m for K (a) R . So we have a complex homomorphism ^ ^ ' R : R m ! K (x): V V Since R m has zero di erential, this yields a map R m ! H (x), and nally, as H (x) = 0, a map ^ : km ! H (x): V As is induced by ', it is a homomorphism of graded k-algebras. Sometimes it will be necessary to use the canonical bases f1 . . . fn of S n, e1 . . . em of S m , and the elements ui = '(ei ) 2 S n . By the choice of ', da(ei ) = ai = dy (ui ). Finally, denotes residue classes mod I . n
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Theorem 2.3.2. With the notation just introduced, (a) 1 : km ! H1 (x) is an isomorphism of k-vector spaces, (b) (I ) = dim ; k H1(Rx),+ dim H (x). (c) 2 (k) = emb dim k 1 2 (Here 2 (k) is the second Betti number of k as an R -module see Section 1.3.) Proof. We constructed such that it maps the canonical basis of km to the homology classes of u1 . . . um. A minimal free resolution of k starts as x R n ;! R ;! 0 and therefore 2 (k) = (Z1(x)). So it is enough for (a), (b), and (c) to prove that dx (f p ^ f q ) 1 p < q n and u i i = 1 . . . m form a minimal system of generators of Z1 (x). Suppose that b 2 Z1(x), b 2 S n . Then dy (b) 2 I , dy (b) = c1 a1 + + cm am = c1dy (u1 ) + + cm dy (um): P Since K (y ) is acyclic, b ; ci ui is a linear combination of the elements dy (fp ^ fq ). That is, the P elements considered P generate Z1(x). Now assume that pq dx(f p ^ f q ) + i u i = 0. We have to show that all the coecients pq , i are in . Lifting the equation to S n gives .
X
^ X iui 2 IS n P iai 2 I . So i 2 by the choice of a. As and applying d yields P I 2 one obtains pq d (fp ^ fq ) 2 2 S n . Looking at the components of the elements d (fp ^ fq ) 2 S n and since y is a minimal system of generators of , one sees that pq 2 for all p, q. m
pq dy (fp fq ) +
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The Koszul complexes with respect to di erent minimal systems of generators of are isomorphic R -algebras see the discussion before 1.6.21. In particular H (x) is essentially independent of x this justi es the notation H (R ) = H (x) and we call H (R ) the Koszul algebra of R . The number "1(R ) = dimk H1 (R ) is the rst deviation of R . It follows immediately from 2.2.5 and 1.6.19 that R is regular if and only if "1 (R ) = 0. So "1(R ) may be considered a measure of how far R deviates from regularity. The following theorem contains the desired intrinsic characterization of complete intersections. m
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Theorem 2.3.3. Let (R k) be a Noetherian local ring. (a) One has "1(R ) = "1 (R^ ). (b) The following are equivalent: (i) R is a complete intersection (ii) "1(R ) = emb ; dim R ; dim R (iii) 2 (k) = 2(k) + 1(k) ; dim R. (c) Suppose that R = S=I with S regular and local. Then R is a complete m
1
intersection if and only if I is generated by an S-sequence.
(a) Choose a minimal system x of generators of . We write x^ for x considered as a sequence in R^ . Then H (R^ ) = H (^x) = H (x) R^ = ^ H (R ) R by 1.6.7. Since H (R ) has nite length, one has H (R ) R^ = H (R ). (b) Because of (a) and the de nition of complete intersection we may assume that R is complete and has a minimal presentation R = S=I . If R is a complete intersection, then there is such a presentation with I generated by an S -sequence, hence "1(R ) = (I ) = dim S ; dim R . Conversely, if "1(R ) = emb dim R ; dim R , then (I ) = dim S ; dim R in an arbitrary minimal presentation, and so I is generated by an S -sequence see 1.6.19. The equivalence of (ii) and (iii) follows immediately from 2.3.2. (c) is proved along the same lines as (b). Proof.
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Permanence properties of complete intersections. As we did for the Cohen{
Macaulay property and regularity we want to discuss how complete intersections behave under certain standard ring extensions. Theorem 2.3.4. Let (R k) be a Noetherian local ring. (a) Suppose x is an R-sequence. Then "1(R=(x)) ; (emb dim R=(x) ; dim R=(x)) = "1 (R ) ; (emb dim R ; dim R ) in particular R is a complete intersection if and only if R=(x) is a complete m
intersection. (b) Suppose R is a residue class ring of a regular local ring. Then if R is a complete intersection, so is R for every Spec R. p
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Proof. Using induction we only need to prove (a) in the case in which x = x R . Suppose rst that x = 2 . Then emb dim R=(x) = emb dim R 1
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2
;
and dim R=(x) = dim R ; 1 furthermore H1 (R ) = H1 (R=(x)) as k-vector spaces by 1.6.13(b). So "1(R ) = "1 (R=(x)). Now suppose that x 2 2 . Then emb dim R=(x) = emb dim R and dim R=(x) = dim R ; 1 moreover we have an exact sequence 0 ;! H1 (R ) ;! H1 (R=(x)) ;! H0 (R ) = k ;! 0 as in the proof of 1.6.16. Thus "1(R=(x)) = "1 (R ) + 1. The proof of (b) is very easy it uses 2.3.3 and basic properties of regular sequences. m
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Remark 2.3.5. Having studied an `abstract' characterization of complete intersections, the reader may expect an `abstract' version of 2.3.4(b) without any restrictions. In fact, such an assertion holds for arbitrary complete intersections as was proved by Avramov 22]. Actually Avramov proved a stronger result, namely the analogue of 2.1.7: suppose (R ) ! (S ) is a at homomorphism of Noetherian local rings then S is a complete intersection if and only if R and S= S are complete intersections. It is not dicult to deduce the localization property from the theorem on at extensions: there is (by faithful atness) a prime ideal R^ such that = \ R the extension R ! R^ is local and at, and R^ is a complete intersection by 2.3.4(b). In 23], Avramov gave quantitatively precise results concerning at extensions (and localizations): let (R ) = "1 (R ) ; (emb dim R ; dim R ) be the complete intersection defect of R then, in the situation of a at extension, (S ) = (R ) + (S= S ). (Limitation of space prevents us including a proof.) Using Avramov's theorem one can also remove the undesirable restrictions in 2.3.6 and 2.3.7 below. In the following we say that a Noetherian ring is a locally complete intersection if all its localizations are complete intersections. Theorem 2.3.6. Let R be a Noetherian ring which is a residue class ring m
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of a regular ring S. Then R is a locally complete intersection if and only if R X1 . . . Xn ] is a locally complete intersection. The same holds for R and R X1 . . . Xn ]].
The proof follows the pattern of that of 2.1.9 one notes that
S X1 . . . Xn] and S X1 . . . Xn]] are regular rings by 2.2.13, and replaces
2.1.7 by Exercise 2.3.20. As with the Cohen{Macaulay property, the argument outlined really proves the stronger local version of 2.3.6: R is a complete intersection if and only if R X1 . . . Xn] is a complete intersection for 2 Spec R , 2 Spec RX1 . . . Xn] with R \ = . A similar remark applies to the following theorem and its proof. Theorem 2.3.7. Let k be a eld, R a Noetherian k-algebra, and K an extension eld of k. Suppose that R is a (ring of fractions of a) nitely p
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generated k-algebra or K is nitely generated as an extension eld. Moreover, suppose that R is a residue class ring of a regular ring. Then R is a locally complete intersection if and only if R k K is a locally complete intersection.
Proof. We saw in 2.1.10 that R k K is a Noetherian ring. Given a prime ideal in R k K , we set = R \ conversely, by faithful atness, for every 2 Spec R there exists 2 Spec R K such that = R \ . Furthermore the extension R ! (R K ) factors through R K so q
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we may replace R by R . By hypothesis, R = S=I with a regular local k-algebra S . Let R be a complete intersection. Then I is generated by a regular sequence g1 . . . gr . Because of faithful atness g1 . . . gr is also a regular sequence in S K . So it suces that S K is a locally complete intersection, and this is immediate from 2.1.11 (in conjunction with 2.2.13). As to the converse, we only do the more dicult case in which K is a nitely generated eld extension. By 2.1.11 again, one has K = (kX1 . . . Xn])T =(h1 . . . hm ) where h1 . . . hm is a regular sequence in (kX1 . . . Xn])T and T is a multiplicatively closed set. Therefore h1 . . . hm is a regular sequence in the faithfully at extension R (kX1 . . . Xn])T = (R X1 . . . Xn])T 0 here T 0 is the image of the natural map kX1 . . . Xn] ! R X1 . . . Xn]. Moreover, (R K ) has the form R X1 . . . Xn] =(h1 . . . hm) with 2 Spec R X1 . . . Xn] such that \ R = and T 0 \ = . By 2.3.4, R X1 . . . Xn ] is a complete intersection. So we can apply the local version of 2.3.6. p
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The Koszul algebra of a complete V intersection. Above we constructed an algebra homomorphism : km H. (R ), m = "1 (R ), starting from a minimal presentation R = S=I , and we saw that 1 : km H1 (R ) is an isomorphism. Such a homomorphism is always present. In fact, H. (R ) is an alternating graded k-algebra therefore, by the universal property
!
!
of the a unique algebra homomorphism V exterior algebra, there exists 0 : H1 (R ) ! H (R ) extending the identity on H1 (R ). Moreover, algeV bra homomorphisms 0 : H1 (R ) ! H (R ) such that 1 andV 01 are V isomorphisms, only di er by the automorphism (01;1 1) of H1 (R ). So we may replace the `abstract' homomorphism 0 by the `concrete' whenever we have a minimal presentation. The situation under consideration can be generalized as follows: S is a ring, I and are ideals generated by sequences a = a1 . . . am and y = y1 . . . yn, and we have I . Then, as above 2.3.2, there is a homomorphism .
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: H. (a S= ) =
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(S= )m ! H (y S=I ): .
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Choose S -free resolutions F of S=I and G of S= . Then there exist complex homomorphisms K (a) ! F and K (y ) ! G . These in turn induce maps : H (a S= ) ;! H (F S= ) = TorS (S=I S= ) : H (y S=I ) ;! H (S=I G ) = TorS (S=I S= ): .
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V
Hence there exist two maps from H (a S= ) = (S= )m to TorS (S=I S= ), namely and . It is crucial that these maps are essentially equal { of course, we must use the proper identi cation of H (F S= ) and H (S=I G ). To this end one forms the double complex F G and considers S=I and S= as complexes concentrated in degree 0. Then one has complex homomorphisms K (a) ! F ! S=I and K (y ) ! G ! S= . Taking tensor products yields a commutative diagram .
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? S= ;;;; K (a) K (y ) ;;;;! S=I K (y ) ? ?
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By a fundamental theorem of homological algebra (318], Theorem 11.21) the bottom row induces an isomorphism H. (F.
= = S= ) ; H (F G ) ;! H (S=I G ): .
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It is this identi cation we need: Lemma 2.3.8. With the notation introduced, s = (;1)s s s . Proof. Let e1 . . . em be a basis of S m and choose elements ui 2 S n with dy (ui) = da (ei ). One has s ( ei ^^ eis ) = u i ^^ u is . Thus it is enough to show that s ( ei ^ ^ eis ) = (;1)s s ( ui ^ ^ u is ). Let z 2 K (a) K (y ) be a cycle. Then the commutativity of the diagram above implies that z , (z ), and (z ) are all mapped to the same homology class. So it suces to exhibit a cycle z with (z ) = ei ^^ eis and (z ) = (;1)s ( ui ^ ^ u is ). Simply take z = (ei 1 ; 1 ui ) (eis 1 ; 1 uis ). In order to see that it is a cycle, one uses the de nition of the di erential of K (a) K (y ) and the fact that a product of cycles is again a cycle. Theorem 2.3.9. Let S be a ring, and a = a1 . . . am and y = y1 . . . yn be Ssequences such that I = (a) = (y ). Then H (y S=I ) = TorS (S=I S= ) V m is (isomorphic with) the exterior algebra (S= ) . Proof. The isomorphism H (y S=I ) = TorS (S=I S= ) results from the fact that K (y ) is a free resolution of S= see 1.6.14. So, with the notation above, is an isomorphism. Similarly is an isomorphism hence , being an algebra homomorphism, is an isomorphism of graded algebras. V (In order to remove the sign in 2.3.8 one would have to replace by (;1).) Corollary 2.3.10. P (a) With the hypotheses of 2.3.9 suppose that m = n. Write ai = ajiyj , i = 1 . . . n. Then I : = I + S = det(aji): (b) In particular, suppose that y is a regular system of parameters in a regular local ring S. Then Soc(S=I ) = (S=I ). 1
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(a) That (I : )=I = HomS (S= S=I ) and Hn (y S=I ) can be identi ed was shown in the proof of 1.6.16. P Now let f1 . . . fn be a basis of S n with dy (fi ) = yi, and set ui = nj=1 aji fj . Then if e1 . . . en is a basis of S n with da(ei ) = ai , one has dy (ui) = ai = da (ei). The theorem implies that Hn (y S=I ) is generated by u 1 ^ ^ un = f 1 ^ ^ f n. So u 1 ^ ^ u n is mapped to the residue class of in S=I by the homomorphism which sends f1 ^^ fn to 1 (and thus gives the identi cation (I : )=I = Hn (y S=I )). (b) By de nition, Soc(S=I ) = (I : )=I . We have completed our preparations for the following beautiful characterization of complete intersections: Theorem 2.3.11 (Tate, Assmus). Let (R k) be a Noetherian local ring. Proof.
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Then the following are equivalent: (a) R is a complete intersection (b) H. (R ) is (isomorphic to) the exterior algebra of H1 (R ) (c) H. (R ) is generated by H1 (R ) (d) H2 (R ) = H1 (R )2.
Here H1 (R )2 is the k-vector space generated by the products w ^ z with w, z 2 H1 (R ). Proof. It was observed in the proof of 2.3.3 that H (R ) is invariant under completion. So we may assume that R is complete and has a minimal presentation R = S=I . The implication (a) ) (b) is a special case of 2.3.9, and (b) ) (c) ) (d) is trivial. For (d) ) (a) we note rst that the map above is an isomorphism. Next, (d) says that 2 : K2 (a) S= ! H2 (y S=I ) = TorS2 (R S= ) is surjective. So 2 is surjective. Choose F as a minimal free resolution of S=I . Then we have a commutative diagram .
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V2 S m ;;;;! S m ;;;;! S ;;;;! 0 ?? ?? y y=
;;;;! F1 ;;;;! S ;;;;! 0 The map 2 is just k, and k being surjective, is surjective itself. F2
It follows immediately that H1 (a) = 0, whence a is an S -sequence by 1.6.19. Theorem 2.3.3 contains a characterization of complete intersections in terms of the numerical invariants dim R , emb dim R = 1(k), and 2(k). It is possible to remove the `non-homological' Krull dimension, and to give a description of complete intersections using 1 (k), 2 (k), and 3(k).
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In order to construct the rst steps in a free resolution of k, we start with the Koszul complex
^2
Rn
;! Rn ;! R ;! 0
unless R is regular, H1 (R ) is non-zero. So we add a free direct summand R m with m = dimk H1 (R ) = "1 (R ), and send its generators e1 . . . em to cycles u1 . . . um whose homology classes generate H1 (R ):
^2
d d ;! R n ;! R ;! 0: The kernel of d2 contains the Koszul cycles d (fi ^ fj ^ fl ) as well as the elements xi ep ; fi ^ up again, f1 . . . fn denotes a basis of R n. In order to
Rm
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`kill' at least these cycles we form the complex T. : (R
with
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n d3
^ ^
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^ ^ ; ^
d3 (fi fj fl ) = dx (fi fj fl ) d3(fi ep ) = xi ep fi up = dx (fi )ep
1
; fi ^ d2(ep):
Part (a) of the following theorem shows that (H2 (T )) is an invariant of R . One writes "2 (R ) = (H2 (T )) and calls this number the second deviation of R . Theorem 2.3.12. Let (R k) be a Noetherian local ring. Then (a) H2 (T ) = H2 (R )=H1(R )2, (b) R is a complete intersection if and only if "2 (R ) = 0, ; ;
(c) "2 (R ) = 3(k) ; 3(k) ; 1 (k) 2(k) ; 2(k) . .
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Proof.
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(a) Let K be the complex .
K. :
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obtained by truncating the Koszul complex K is a subcomplex of T . The quotient T =K is isomorphic to .
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L. : R n
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with R n R m in degree 3. Consider the exact sequence of homology H3 (L ) ! H2 (K ) ! H2 (T ) ! H2 (L ) ! H1 (K ) ! H1 (T ) = 0: .
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The map H2 (L ) ! H1 (K ) is an isomorphism since both vector spaces have dimension m. Hence we have an exact sequence H3 (L ) ! H2 (K ) ! H2 (T ) ! 0. The kernel of R n R m ! R m is obviously generated by the elements dx (fi ^ fj ) eq and up eq . An analysis of the connecting homomorphism shows that the class of dx (fi ^ fj ) eq goes to that of dx(fi ^ fj ) ^ uq which is a boundary in the Koszul complex (see the formulas above 1.6.4) the class of up eq goes to that of up ^ uq whence the image of H3 (L ) ! H2 (K ) is H1 (K )2. (b) follows immediately from (a) and 2.3.11. (c) The veri cation is similar to that of 2.3.2(c), and therefore left to the reader. Remark 2.3.13. The equation d3(fi ep) = dx (fi )ep ; fi ^ d2(ep ) suggests that d3 is a component of an antiderivation of an alternating algebra. In fact, the choice of d3 is part of Tate's construction of resolutions with algebra structures 369]. Suppose that A is an alternating graded R -algebra equipped with an antiderivation @ of degree ;1 such that @2 = 0, and consider the homology H (A ) = Ker @= Im @. Let z 2 Hp (A ) be a non-zero homology element. Then one may adjoin a variable to `kill' the cycle z representing z : (i) If p is even, let B be the exterior algebra in a variable of degree p + 1, i.e. B = R Re with R in degree 0 and Re = R in degree p + 1, the multiplication being de ned by e2 = 0. (ii) For p odd let B be theL`divided power algebra' over R in a variable of degree p + 1, i.e. B = 1j =0 Rej with Rej = R in degree j (p + 1), the multiplication being de ned by ej el = ((j + l )!=j ! l !)ej +l. In both cases A B is again an alternating algebra, and there is a unique antiderivation d on A B such that d jA 1 = @, d (e) = z in case (i), and d (ej ) = zej ;1 for all j in case (ii) moreover, one has d 2 = 0. It follows easily that Hq (A B ) = Hq (A ) for q < p and Hp (A B ) = Hp (A )=R z. In order to resolve the residue class eld k of a local ring (R k) one starts with the R -algebra T (0) = R . Let "0(R ) = emb dim R = ( ), and successively adjoin "0 (R ) variables of degree 1 `to kill the zero-cycles'. The resulting algebra T (1) is the Koszul complex of a minimal system of generators of . Next one adjoins "1 (R ) variables of degree 2 to kill "1(R ) cycles generating H1 (T (1)). The algebra T (2) thus constructed has H1 (T (2)) = 0. It is a theorem of Tate 369] that in the case of a complete intersection the complex T (2) is a minimal free resolution of k. However, if R is not a complete intersection, then one has to adjoin "2 (R ) variables of degree 3 etc. A famous theorem of Gulliksen 146] and Schoeller 330] says that the resolution of k obtained in this way is always minimal. For a comprehensive study of resolutions with an algebra structure we refer the reader to Gulliksen and Levin 147]. .
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2.3. Complete intersections
As above, let (R k) be a Noetherian local ring. V We saw in 2.3.11 that surjectivity of the natural homomorphism : H1 (R ) ! H (R ) is already sucient for R to be a complete intersection. This is also true for injectivity, at least when R contains a eld. Theorem 2.3.14. Let (R k) be a Noetherian local ring containing a eld. m
.
m
Then (a) H1 (R )j = 0 for j > emb dim R dim R (b) in particular, R is a complete intersection if (and only if) the natural V map : H1 (R ) H. (R ) is injective.
;
!
Proof. It is harmless to complete R so that we may assume that R has a minimal presentation R = S=I as above. In order to prove (a) we must anticipate Corollary 9.5.3: it says that, with the notation of 2.3.8, j = 0 for j > emb dim R ; dim R . Since, in the present circumstances, j is an isomorphism, one has j = 0 for j > emb dim R ; dim R . As V H1 (R )j = j ( j H1 (R )), one has H1 (R )j = 0. This proves (a), and (b) is an obvious consequence of (a).
The restriction to local rings containing a eld is forced upon us since there does not yet exist a proof of 9.5.3 without this restriction. However, one always has H1 (R )j = 0 for j > emb dim R ; dim R + 1 so that the gap in 2.3.14 is as small as it could be see 9.5.7. The reader may have noticed that 2.3.14 is trivial for Cohen{Macaulay rings R : if j > emb dim R ; depth R , then even Hj (R ) = 0 by 1.6.17. On the other hand, for Cohen{Macaulay R the non-vanishing of H1 (R )p for p = emb dim R ; dim R conveys the strongest possible information: R is a complete intersection. More generally, we have the following theorem. Theorem 2.3.15. Let (R k) be a Noetherian local ring. Then R is a complete intersection if (and only if) H1 (R )p 6= 0 for p = emb dim R ; depth R. Proof. By virtue of 2.3.14 the hypothesis H1 (R )p 6= 0 for p = emb dim R ; depth R forces R to be Cohen{Macaulay if it contains a eld. Because of this restriction we give a proof not using 2.3.14. First we reduce to the case depth R = 0. So suppose that depth R > 0. Then there exists an x 2 n 2 which is not a zero-divisor, and 1.6.13 furnishes us with an isomorphism : H (R ) = H (R 0 ), R 0 = R=(x). It is not dicult to verify that is a k -algebra isomorphism after all, is induced V by , being the composition R n ! R n;1 ! (R 0)n;1 (see the proof of 1.6.12). Furthermore emb dim R 0 ; depth R 0 = emb dim R ; depth R , and R is a complete intersection if and only if this holds for R 0 . It remains to show that R is a zero dimensional complete intersection if H1 (R )n 6= 0 for n = emb dim R . The complex K (R ) has length n so m
m
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Bn+1(R ) = 0, and therefore
H1 (R )n = (Z1 (R )n + Bn+1(R )) Bn+1(R ) = Z1 (R )n:
V
Consider an exact sequence R m 2 R n ;! R nP;! ;! 0 as above. Choose elements v1 . . . vn 2 Im d2 = Z1(R ), vi = vji fj where f1 . . . fn is a basis of R n . Then v1 ^^ vn = det(vji )f1 ^^ fn, whence H1 (R )n 6= 0 is equivalent to In (d2) 6= 0. It remains to apply the next theorem. Theorem 2.3.16 (Wiebe). Let (R k) be a Noetherian local ring, and ' R r ;! R n ! ! 0 a presentation of its maximal ideal. If In (') 6= 0, then R is a complete intersection of dimension zero (and conversely). Proof. The ideal In(') is the zeroth Fitting ideal of , which is an invariant of . Therefore it is enough to consider a special presentation. Moreover, we may assume that R is complete. Then R = S=I where (S ) is a regular local ring and I 2 . Let y = y1 . . . yn be a regular system of parametersPof S , and a = a1 . . . am a minimal system of generators of I . Write ai = ajiyj . The converse of the theorem is part of Corollary 2.3.10 it implies the following claim which is crucial in what follows. Let b = b1 . . . bn be a maximal S-sequence, and J 0 an ideal properly containing P J = (b) then det(bji ) 2 J 0 where the bji are chosen such that bi = bjiyj . In fact, det(bji )S=J is the socle of S=J . Since it has dimension 1 over k, it is contained in every non-zero ideal of S=J . Let f1V . . . fn be a basis of S n , and e1V . . . em a basis of S m . De ne 2 m n n : S SP ! S by the Koszul map 2 S n ! S n with respect to y and (ei ) = ajifj . We saw in 2.3.2 that Coker S=I = . So the theorem claims that In() I unless S=I is a zero dimensional complete intersection. Choose an n n submatrix U of (a matrix of) . If U involves a column corresponding to one of the elements fi ^ fj , then det U 2 I , since, on the level of R , we are taking the exterior product of n cycles at least one of which is a boundary: (det U )f 1 ^ ^ f n 2 Bn+1(R ) = 0. Therefore it is enough to consider submatrices U of (aji). For simplicity of notation one may assume U consists of the rst n columns. If a1 . . . an is a regular sequence, then I contains J = (a1 . . . an) properly since S=I is not a complete intersection. So det U I by the claim above. If a1 . . . an is not a regular sequence, then dim S=J > 0, and it is certainly enough to show that det U 2 J , that is, we may assume that I = (a1 . . . an ). We will show that det U 2 I + p for all p 2 N. Then det U 2 I follows from Krull's intersection theorem. Fix p 2 N. According to Exercise 2.3.17 one nds elements a00i 2 p+1 such that a0i = ai + a00i , i = 1 . . . n, is a d2
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Notes
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P
regular sequence. Write a0i = a0jiyj then aji ; a0ji 2 p follows from the quasi-regularity of the regular sequence y , in other words, from the fact that the associated graded ring gr (S ) is a polynomial ring (see 1.1.8). Therefore det U ; det(a0ji) 2 p : n
n
n
The ideal I + p properly contains (a01 . . . a0n) I + p+1. Once more the auxiliary claim above is applied, and it yields det(a0ji) 2 I + p . n
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n
Exercises
Let (R ) be a Noetherian local ring of depth t, and a1 . . . at 2 . Then, given p 2 N, show there exist a1 . . . at 2 p such that a1 + a1 . . . at + at is an R -sequence. 2.3.18. Let S be a regular local ring of dimension 4, and y1 . . . y4 a regular system of parameters. Let I = (y1 y2 y3 y4 y1 y3 + y2 y4 ) and R = S=I . (a) Construct a minimal free resolution of R . (b) Prove depth R = 0 and dim R = 2. (c) Show that the vector space H1 (R )2 has dimension 3, the maximal value for an ideal generated by 3 elements, but R is not a complete intersection. 2.3.19. Prove all the claims in the second paragraph of 2.3.13. 2.3.20. Let ' : (S1 1 ) ! (S2 2 ) be a at local homomorphism of regular rings, and I S1 an ideal. Verify S1 =I is a complete intersection if and only if S2 =IS2 is a complete intersection. 2.3.21. Let R be a Noetherian graded ring. Show: (a) For 2 Spec R the localization R is a complete intersection if and only if R is. (b) The following are equivalent: (i) R is locally a complete intersection (ii) R is a complete intersection for all graded prime ideals (iii) R( ) is locally a complete intersection for all graded prime ideals . (c) Suppose in addition that (R ) is local. Then R is locally a complete intersection if and only if R is a complete intersection. Hint: 1.6.33. 2.3.22. Extend 2.3.2 and 2.3.12 to the following theorem which Serre 332] used to prove 2.2.7(c) ) (a): let (R k) be a Noetherian local ring, and x a minimal system of generators of then the natural map K. (x k) ! TorR. (k k) (see 1.6.9) is injective. 2.3.17.
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Notes The origins of the theory of Cohen{Macaulay rings are the unmixedness theorems of Macaulay 263] and Cohen 71] and the notion of perfect ideals, which also goes back to Macaulay and was clari ed by
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2. Cohen{Macaulay rings
Grobner 141]. The present shape of the theory was formed by Auslander and Buchsbaum 18], Nagata 283], and Rees 303]. It seems that Cohen{Macaulay modules made their rst appearance in Auslander and Buchsbaum 19]. The characterization 2.1.27 of graded Cohen{Macaulay rings is essentially due to Hochster and Ratli 200] and Matijevic and Roberts 268]. By analogy to the desingularization, one can try to `Macaulayfy' a Noetherian scheme. Special results in this direction were obtained by Brodmann 51] and Faltings 100]. Recently Kawasaki 234] has proved a general theorem on the existence of `Macaulay cations'. Of all the notions generalizing Cohen{Macaulay rings and modules, the concept of Buchsbaum ring or module is the most important see Stuckrad and Vogel 365] and Schenzel 329]. The `classical' theory of regular local rings, to be found in Zariski and Samuel 397], Vol. II, was developed by Krull 242], Chevalley 68], Cohen 71], and Zariski 396]. It depends in an essential way on power series methods, and is therefore mainly restricted to local rings containing a eld. The problems it could not solve were (i) the regularity of a localization of a regular local ring R (even if R contains a eld), and (ii) the factoriality of such rings (because of the Cohen structure theorem this is easy if R contains a eld). The breakthrough was the theorem 2.2.7 of Auslander and Buchsbaum 17], 18] and Serre 332] which not only solved the localization problem: `this resounding triumph of the new homological method marked a turning point of the subject of commutative Noetherian rings' (Kaplansky 230], p. 159). Theorem 2.2.8 was independently given by Ferrand 105] and Vasconcelos 379] it generalizes Kaplansky's proof of 2.2.14(c) ) (a) (see 270], x19). The problem of factoriality was solved by Auslander and Buchsbaum 20], using results of Zariski and Nagata who reduced the theorem to the case of Krull dimension 3. See Nagata 284], p. 217 for a minute history. The proof we have reproduced is due to Kaplansky (except for the application of the Hilbert{Burch theorem). That regular local rings are factorial can be expressed by saying that every ideal I has a greatest common divisor: there is a regular element a and an ideal J of grade 2 such that I = aJ . MacRae 265] proved this fact for every ideal with a nite free resolution. Another (related) generalization is that every module with a nite free resolution over a normal domain has divisor class zero see 47], Ch. VII, x4. The most concrete and computationally e ective result is the factorization theorem of Buchsbaum and Eisenbud 64]. The notion of complete intersection is classical in algebraic geometry. An abstract de nition in terms of local algebra was given by Scheja
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Notes
322], together with 2.3.3. Our de nition is that of Grothendieck 142]. Avramov's contributions 22], 23] have been described in 2.3.5 they ultimately justi ed the abstract notion of complete intersection. The program of Tate's seminal paper 369] has been outlined in Remark 2.3.13. Assmus 14] used Tate's method to give the description 2.3.11 of complete intersections in terms of their Koszul algebras. There are several papers devoted to the characterization of complete intersections by the vanishing of a deviation "i (which we de ned only for i = 1 2) the question was nally settled by Halperin 148] who showed that "i > 0 for all i if R is not a complete intersection. Wiebe's theorem 2.3.16 appeared in 395] see Kunz 244], Hilfssatz 1, for a related result. A driving force in this area of researchP was the problem (posed by Serre 334]) of whether the Poincare series i (k)ti of a Noetherian local ring (R k) is a rational function of t. After several special cases had been solved positively, the general question was answered negatively by Anick 9]. In several theorems we studied the behaviour of ring-theoretic properties under at extensions R ! S . Avramov, Foxby, and Halperin 31] investigated the more general situation in which S is supposed of nite at dimension over R . As we have seen, another `homologically nice' type of extension is that of passing to a residue class modulo a regular sequence. Avramov and Foxby have essentially completed a program which aims at the uni cation of these types of extensions by introducing a suitable notion of bre see 27], 28], 29], 30]. m
3 The canonical module. Gorenstein rings
The concept of a canonical module is of fundamental importance in the study of Cohen{Macaulay local rings. The purpose of this chapter is to introduce the canonical module and derive its basic properties. By de nition it is a maximal Cohen{Macaulay module of type 1 and of nite injective dimension. In the rst two sections we investigate the injective dimension of a module, and prove Matlis duality which plays a central role in Grothendieck's local duality theorem. Actually the canonical module has its origin in this theory. Here the canonical module is introduced independently of local cohomology which is an important notion in itself and will be treated later in this chapter. A ring which is its own canonical module is called a Gorenstein ring. Next to regular rings and complete intersections, Gorenstein rings are in many ways the `nicest' rings. Distinguished by the fact that they are of nite injective dimension, they have various symmetry properties, as reected in their free resolution, their Koszul homology, and their Hilbert function. The last aspect will be discussed in the next chapter. Gorenstein rings of embedding dimension at most two are complete intersections. The rst non-trivial Gorenstein rings occur in embedding dimension three, and they are classi ed by the Buchsbaum{Eisenbud structure theorem. In the nal section the canonical module of a graded ring is introduced. 3.1 Finite modules of nite injective dimension In this section we study injective resolutions of nite modules. We shall see that the injective dimension of a nite module M over a Noetherian local ring R either is in nite or equals the depth of R , and is bounded below by the dimension of M . Thus, quite contrary to the behaviour of projective dimension, the injective dimension, if it is nite, does not depend on the module. We introduce Gorenstein rings and show that Gorenstein rings are Cohen{Macaulay rings. De nition 3.1.1. Let R be a ring. An R -module I is injective if the functor HomR ( I ) is exact. 88
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3.1. Finite modules of nite injective dimension
Notice that HomR ( I ) is always left exact. Thus the R -module I is injective if and only if HomR ( I ) is right exact as well. We now list some useful characterizations of injective modules. Proposition 3.1.2. Let R be a ring and I an R-module. The following
conditions are equivalent: (a) I is injective (b) given a monomorphism ' : N M of R-modules, and a homomorphism : N I, there exists a homomorphism : M I such that = ' (c) given R-modules N M, and a homomorphism : N I, there exists a homomorphism : M I such that N = in other words, : N I can be extended to a homomorphism : M I (d) for all ideals J R, every homomorphism J I can be extended to R, that is, Ext1R (R=J I ) = 0 (e) let M be an R-module with I M then I is a direct summand of M (f) Ext1R (M I ) = 0 for all R-modules M (g) ExtiR (M I ) = 0 for all R-modules M and all i > 0.
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The ExtiR ( I ) are the right derived functors of HomR ( I ). The equivalence of (a), (f) and (g) follows therefore from the general properties of right derived functors. For details we refer to 318], Section 6. (a) ) (b): The monomorphism ' : N ! M induces the homomorphism Hom(' I ): HomR (M I ) ;! HomR (N I ) where Hom(' I )( ) = ' for all 2 HomR (M I ). By assumption, Hom(' I ) is an epimorphism, and so 2 HomR (N I ) is of the form ' for some 2 HomR (M I ). Similarly one proves (b) ) (a). The implications (b) () (c) and (c) ) (d) are clear. (d) ) (c): We consider the set of all pairs (U '), where U is a submodule of M with N U , and where ' extends . We order this set partially: (U1 '1) (U2 '2) if and only if U1 U2 and '1 = '2jU . By Zorn's lemma there exists a maximal element (U 0 '0) in this set. Suppose U 0 6= M then we may choose x 2 M n U 0 . Set W = U 0 + Rx then W =U 0 = R=J for some ideal J in R . Applying the functor HomR ( I ) to the exact sequence 0 ;! U 0 ;! W ;! R=J ;! 0 we obtain the exact sequence HomR (W I ) ;! HomR (U 0 I ) ;! Ext1R (R=J I ): Since by assumption Ext1R (R=J I ) = 0, it follows from the exact sequence that any homomorphism from U 0 to I can be extended to a homomorphism W ! I , contradicting the maximality of (U 0 '0 ). Thus we have shown that U 0 = M . Proof.
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(c) ) (e): There exists a homomorphism : M ! I with jI = idI . Therefore, M = I Ker : (e) ) (b): Given a monomorphism ' : N ! M and a homomorphism : N ! I , we want to nd : M ! I such that = '. In order to do this, we construct a commutative diagram ' ;;;;! M ? ?
N
?y
I
?y
;;;;! W
where is injective. In fact, we may choose W = (M I )=C with C = f('(x) ; (x)): x 2 N g and are the natural homomorphisms arising from this situation. (This diagram is called the pushout of and '.) Since is injective by construction, it is split injective by our assumption (e). This means that there exists a homomorphism : W ! I with = idI . The homomorphism : M ! I , = , is the desired extension of . Corollary 3.1.3. Let R be a Noetherian ring. (a) If I is an injective R-module and S is a multiplicatively closed set of R, then IS is an injective RS -module. (b) If (I ) 2 is a family of injective R-modules, then the direct sum I=
M 2
I
is an injective R-module. Proof. (a) Let J be an ideal of R . Since R is Noetherian one has Ext1RS (RS =JRS IS ) = Ext1R (R=J I )S = 0: Since every ideal of RS is extended from R , 3.1.2 yields that IS is an injective RS -module. (b) By 3.1.2 it is L enough to show that for an ideal J of R , any homomorphism ' : J 2 I extends to R . Since J is nitely generated Ln I . there exists a nite subset 1 . . . n of such that Im ' i=1 i We denote by 'j the j -th component of '. Since I i is injective we can extend 'L I i to a homomorphism I i . It is clear that i : J Pn i(a), ai : RR, extends : R ' to all of R . 2 I with (a) = i=1
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Remark 3.1.4. It is a simple exercise to see that for an arbitrary ring R any direct product of injective modules is injective. It is however essential to require that R is Noetherian (as we have done in 3.1.3) to obtain a similar result for direct sums. In fact, this property characterizes Noetherian rings see 318], Theorem 4.10.
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3.1. Finite modules of nite injective dimension
An R -module M is divisible if for every regular element r 2 R , and every element m 2 M , there exists an element m0 2 M such that m = rm0 . Condition 3.1.2(d) has the following consequence. Corollary 3.1.5. Let R be a ring and I an R-module. (a) If I is injective, then I is divisible. (b) If R is a principal domain and I is divisible, then I is injective. Proof. The property that I is divisible is equivalent to the property that every homomorphism : (r) ! I , r regular, can be extended to R . Therefore (a) and (b) follow from 3.1.2(d). For later applications we note the following result about change of rings. Lemma 3.1.6. Let ' : R ! S be a ring homomorphism, and let I be an injective R-module. Then HomR (S I ) (equipped with the natural S-module structure) is an injective S-module. Proof. Let M be an S -module. There is a natural isomorphism HomS (M HomR (S I )) = HomR (M I ) of S -modules. Indeed, to 2 HomS (M HomR (S I )) one assigns 0 2 HomR (M I ) where 0 (x) = (x)(1) for all x 2 M . Thus the exactness of the functor HomR ( I ) on the category of S -modules (considered as R modules via ') implies the exactness of the functor HomS ( HomR (S I )). This means that HomR (S I ) is an injective S -module. De nition 3.1.7. Let R be a ring and M an R -module. A complex I : 0 ;! I 0 ;! I 1 ;! I 2 ;! with injective modules I i is an injective resolution of M if H 0 (I ) =M and H i (I ) = 0 for i > 0. While it is obvious that every module has a projective resolution, it is less obvious that it has an injective resolution. It is however clear that an injective resolution can be constructed by resorting to the following result. Theorem 3.1.8. Let R be a ring. Every R-module can be embedded into an .
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injective R-module.
The Z-module Q is divisible, and hence injective. Therefore any free Z-module F can be embedded into an injective Z-module I . We just take suciently many copies of Q. If G is an arbitrary Z-module, then G = F=U , and we can embed G into I=U . It is immediate that I=U is again divisible, and hence injective. Thus the theorem is proved for Z-modules. Proof.
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Now let R be an arbitrary ring, and let M be an R -module. The map : M HomZ (R M ) with (x)(a) = ax for all x M and all a R is an R -module monomorphism. By our considerations above, the R -module M can be embedded as a Z-module into an injective Z-module I . This
!
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inclusion induces a monomorphism : HomZ (R M ) ;! HomZ (R I ): By 3.1.6, the R -module J = HomZ (R I ) is injective, and thus : M ! J is the desired embedding.
Injective dimension. Let R be a ring and M an R -module. The injective dimension of M (denoted inj dim M or inj dimR M ) is the smallest integer n for which there exists an injective resolution I . of M with I m = 0 for m > n. If there is no such n, the injective dimension of M is in nite.
The following observation is an immediate consequence of 3.1.3 and the exactness of localization. Proposition 3.1.9. Let R be a Noetherian ring, M an R-module and S a multiplicatively closed set. Then inj dimRS MS inj dimR M. In the next proposition we characterize the injective dimension of a module homologically. Proposition 3.1.10. Let R be a ring and M an R-module. The following conditions are equivalent: (a) inj dim M n (b) ExtnR+1(N M ) = 0 for all R-modules N (c) ExtRn+1(R=J M ) = 0 for all ideals J of R.
Proof. (a)
) (b) follows from the fact that ExtnR+1(N M) can be computed
from an injective resolution of M . (b) ) (c) is trivial. (c) ) (a): Let 0 ;! M ;! I 0 ;! I 1 ;! ;! I n;1 ;! C ;! 0 be an exact sequence, where the modules I j are injective. From the fact that ExtiR (R=J I ) = 0 for i > 0 if I is an injective R -module, the above exact sequence yields the isomorphism Ext1R (R=J C ) = ExtnR+1(R=J M ) and so Ext1R (R=J C ) = 0 for all ideals J of R . This is condition (d) of 3.1.2, and so C is injective. Proposition 3.1.10 can be sharpened if R is Noetherian. We rst observe:
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Lemma 3.1.11. Let R be a Noetherian ring, M an R-module, N a nite R-module and n > 0 an integer. Suppose that ExtnR (R= M ) = 0 for all 2 Supp N. Then ExtnR (N M) = 0. Proof. N has a nite ltration whose factors are isomorphic to R= for certain 2 Supp N . Hence the lemma follows from the `additivity' of the vanishing of ExtnR ( M ). Corollary 3.1.12. Let R be Noetherian and M an R-module. The following p
p
p
p
conditions are equivalent: (a) inj dim M n (b) ExtnR+1(R= M ) = 0 for all
p
p
2 Spec R.
Lemma 3.1.11 has another remarkable consequence. Proposition 3.1.13. Let (R k) be a Noetherian local ring, a prime ideal dierent from , and M a nite R-module. If ExtnR+1(R= M ) = 0 for all prime ideals 2 V ( ), 6= , then ExtnR (R= M ) = 0. Proof. We choose an element x 2 n . The element is R= -regular, and therefore we get the exact sequence m
p
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q
q
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0 ;! R= ;! R= which induces the exact sequence x
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p
;! R=(x ) ;! 0 p
ExtnR (R= M ) ;! ExtnR (R= M ) ;! ExtnR+1(R=(x ) M ): Since V (x ) f 2 V ( ): 6= g, Lemma 3.1.11 and our assumption imply ExtnR+1(R=(x ) M ) = 0 so that multiplication by x on the nite R -module ExtnR (R= M ) is a surjective homomorphism. The desired result follows from Nakayama's lemma. It is now easy to derive the following useful formula for the injective dimension of a nite module. Proposition 3.1.14. Let (R k) be a Noetherian local ring, and M a nite x
p
p
p
q
p
q
p
p
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p
R-module. Then
m
inj dim M = supfi : ExtiR (k M ) 6= 0g:
We set t = supfi : ExtiR (k M ) 6= 0g. It is clear that inj dim M t. To prove the converse inequality, note that the repeated application of 3.1.13 yields ExtiR (R= M ) = 0 for all 2 Spec R and all i > t. According to 3.1.12 this implies inj dim M t.
Proof.
p
p
94
3. The canonical module. Gorenstein rings
Corollary 3.1.15. Let (R k) be a Noetherian local ring and M a nite
R-module. If x
2
m
m
is an element which is R- and M-regular, then
inj dimR=(x) M=xM = inj dimR M ; 1:
The proof is an immediate consequence of 3.1.14 and the following result of Rees 302], Theorem 2.1. Lemma 3.1.16. Let R be a ring, and let M and N be R-modules. If x is an R- and M-regular element with x N = 0, then for all i
0.
ExtiR+1(N M ) = ExtiR=(x)(N M=xM )
Proof. We set R i +1 ExtR ( M ), i
= M=xM , and show that the functors = R=(x) and M 0, from the category of R -modules into itself are the ). To see this, we have to verify right derived functors of HomR ( M i +1 (1) the functors ExtR ( M ), i 0, are strongly connected, ) are equivalent, (2) the functors Ext1R ( M ) and HomR ( M i +1 (3) ExtR (F M ) = 0 for all i > 0 and every free R -module F . (An axiomatic description of the Ext groups as functors in the second variable is given in 318], Theorem 7.22. Similarly the Ext groups can be described axiomatically as functors in the rst variable see 318], Exercise 7.27.) x ;! 0 (1) is obvious. The exact sequence 0 ;! M ;! M ;! M yields the exact sequence x HomR (N M ) ;! HomR (N M ) ;! Ext1R (N M ) ;!
Since HomR (N M ) = 0, and since x annihilates Ext1R (N M ), we obtain the natural isomorphism ) HomR (N M = Ext1R (N M ): This proves (2). Finally, (3) is clear since proj dimR F 1 for every free R -module F . We now present the main result of this section. Theorem 3.1.17. Let (R k) be a Noetherian local ring, and let M be a m
nite R-module of nite injective dimension. Then
dim M inj dim M = depth R:
95
3.1. Finite modules of nite injective dimension
Let 0 1 d = be a maximal chain of prime ideals in Supp M . We show by induction on i that ExtiR i (k( i ) M i ) 6= 0. In particular, it will follow that ExtdR (k M ) 6= 0 for d = dim M , so that dim M inj dim M by 3.1.14. If i = 0, then 0R 2 Ass M , and therefore HomR (k( 0 ) M ) 6= 0. Now suppose i > 0. We set B = R i then Proof.
p
p
p
m
p
p
p
p
p
0
p
0
p
0
p
p
0
ExtiB;1 (B= i;1B M i ) i; = ExtiR;1i; (k( i;1) M i; ) 6= 0 p
p
p
p
1
1
p
p
p
1
by the induction hypothesis, and so ExtiB;1(B= i;1B M i ) 6= 0. It follows from 3.1.13 that ExtiB (k( i ) M i ) 6= 0: To prove the equality inj dim M = depth R , we set r = inj dim M and t = depth R . Let x = x1 . . . xt be a maximal R -sequence. Then the Koszul complex K (x) is a minimal free resolution of R=(x) by 1.6.19 so that proj dim R=(x) = t and furthermore ExttR (R=(x) M ) is isomorphic to the t-th Koszul cohomology H t (x M ). It follows from 1.6.10 that H t (x M ) = H0 (x M ) = M=xM 6= 0. This implies r t. On the other hand, since depth R=(x) = 0, there is an embedding k ! R=(x) which induces an epimorphism ExtrR (R=(x) M ) ;! ExtrR (k M ) p
p
p
p
.
since ExtrR+1(N M ) = 0 for all R -modules N . But ExtrR (k M ) 6= 0 by 3.1.14, and so ExtrR (R=(x) M ) 6= 0. It follows that t = proj dimR R=(x) r. Gorenstein rings. We are now going to introduce an important class of
local rings. As for regular rings, this class can be characterized in terms of homological algebra. De nition 3.1.18. A Noetherian local ring R is a Gorenstein ring if inj dimR R < 1. A Noetherian ring is a Gorenstein ring if its localization at every maximal ideal is a Gorenstein local ring. The Gorenstein property is stable under standard ring operations. To begin with we show Proposition 3.1.19. Let R be a Noetherian ring. (a) Suppose R is Gorenstein. Then for every multiplicatively closed set S in R the localized ring RS is also Gorenstein. In particular, R is Gorenstein for every Spec R. (b) Suppose x is an R-regular sequence. If R is Gorenstein, then so is R=(x). The converse holds when R is local. (c) Suppose R is local. Then R is Gorenstein if and only if its completion R^ is Gorenstein. p
2
p
96
3. The canonical module. Gorenstein rings
(a) Let be a maximal ideal of RS . The ideal is the extension of a prime ideal in R , and so (RS ) = R . Let be a maximal ideal of R containing . Then R is a localization of the Gorenstein local ring R . From 3.1.9 the conclusion follows. (b) Without restriction we may assume that R is local. Thus (b) is an immediate consequence of 3.1.15. (c) Let k be the residue eld of R . Use that ExtiR (k R )b = ExtiR^ (k R^ ). Proof.
q
q
p
m
q
p
p
p
m
In concluding this section we clarify the position of the Gorenstein rings in the hierarchy of Noetherian local rings. Proposition 3.1.20. Let (R k) be a Noetherian local ring. Then we have m
the following implications: R is regular R is a complete intersection R is Cohen{Macaulay:
)
)
) R is Gorenstein
The rst implication is trivial. If R is regular, then its global homological dimension is nite (see 2.2.7), and hence ExtiR (k R ) = 0 for i 0. It follows from 3.1.14 that R is Gorenstein. In view of 3.1.19(c) we may as well assume that R is complete. Now 3.1.19(b) implies that a complete intersection is Gorenstein. The last implication follows from 3.1.17. All the implications of 3.1.20 are strict. This is clear for the rst, and will be shown for the other implications in the next section (see 3.2.11), where we derive a di erent, more easily veri able, characterization of Gorenstein rings. Proof.
Exercises 3.1.21. Let R be a principal ideal domain with eld of fractions K . Prove that 0 ! K ! K=R ! 0 is an injective resolution of R . 3.1.22. Let k be a eld, and let R be a local k-algebra of nite k-dimension. Show that the R -module Homk (R k) is an indecomposable (see the de nition before 3.2.6) injective R -module. 3.1.23. Let R be a Noetherian local ring. If there exists a non-zero nite injective R -module, then deduce R is Artinian. 3.1.24. Let (R k) be a Noetherian local ring, M 6= 0 and N 6= 0 nite R -modules. If inj dim N < 1, then deduce the following result of Ischebeck 226]: depth R ; depth M = supfi : ExtiR (M N ) 6= 0g: In particular, if R admits a nite module of nite injective dimension, then show that the depth of any nite R -module does not exceed the depth of R . (Bass' conjecture claims more: in the above situation R is Cohen{Macaulay. In 9.6.2 a proof of this conjecture will be given, provided R contains a eld.) m
97
3.2. Injective hulls. Matlis duality
Let R be a Gorenstein local ring, and M a nite R -module. Show proj dim M < 1 if and only if inj dim M < 1. (Foxby 114] proved the following remarkable characterization of Gorenstein rings: if a Noetherian local ring possesses a nite module M for which inj dim M < 1 and proj dim M < 1, then it is Gorenstein.) 3.1.26. Let (R k) be a Noetherian local ring. If inj dim k < 1, show R is regular. 3.1.25.
m
3.2 Injective hulls. Matlis duality We saw in Section 3.1 that any module M can be embedded into an injective module. Here we will show that such an embedding can be chosen minimal. In this case the corresponding injective module is unique up to isomorphism, and is called the injective hull of M . We will see that for a Noetherian ring R an injective module can be uniquely written as a direct sum of indecomposable injective modules, and the indecomposable injective R -modules are just the injective hulls of the cyclic R -modules R= , where 2 Spec R . If (R k) is a complete Noetherian local ring, and E is the injective hull of k, then the functor HomR ( E ) establishes an anti-equivalence between the category of Artinian R -modules and the category of nite R -modules. This result is known as the main theorem of Matlis duality. De nition 3.2.1. Let R be a ring and let N M be R -modules. M is an essential extension of N if for any non-zero R -submodule U of M one has U \ N 6= 0. An essential extension M of N is called proper if N 6= M . The following proposition gives a new characterization of injective modules. Proposition 3.2.2. Let R be a ring. An R-module N is injective if and only p
p
m
if it has no proper essential extension.
Let N M be an extension. If N is injective, then N is a direct summand of M . Let W be a complement of N in M . Then N \ W = 0, and so, if the extension is essential, W = 0. It follows that N = M . Conversely, suppose that N has no proper essential extension. Given a monomorphism ' : U ! V and a homomorphism : U ! N , we want to construct : V ! N such that = '. As in the proof of 3.1.2 we consider the pushout diagram Proof.
' ;;;;! V ?? ??
U
y
N
y
;;;;! W
98
3. The canonical module. Gorenstein rings
Here is a monomorphism, since ' is a monomorphism. Thus we may consider N as a submodule of W . Employing Zorn's lemma one shows that there exists a maximal submodule D W such that N \ D = 0, and so N may even be considered as a submodule of W =D obviously, W =D is an essential extension of N . It follows that N = W =D, since N has no proper essential extension, and so W = N D. Let : W ! N be the natural projection of W onto the rst summand. The composition : V ! N is an extension of . De nition 3.2.3. Let R be a ring and M an R -module. An injective module E such that M E is an essential extension is called an injective hull of M . Our notation will be E (M ) or ER (M ). The next proposition justi es this name. Proposition 3.2.4. Let R be a ring and M an R-module. (a) M admits an injective hull. Moreover, if M I and I is injective, then a maximal essential extension of M in I is an injective hull of M. (b) Let E be an injective hull of M, let I be an injective R-module, and : M I be a monomorphism. Then there exists a monomorphism ' : E I such that the diagram M E
!
?? ;;;;!
y = '
!
I is commutative, where M E is the inclusion map. In other words, the injective hulls of M are the `minimal' injective modules in which M can be embedded. (c) If E and E 0 are injective hulls of M, then there exists an isomorphism ' : E E 0 such that the diagram
!
!
M
' JJ^ E ;;;! E 0 commutes. Here M ! E and M ! E 0 are the inclusion maps. Proof. (a) We embed M into an injective R -module I . Consider the set S of all essential extensions M N with N I . Zorn's lemma applied to S yields the existence of a maximal essential extension M E with E I . We claim that E has no proper essential extension, and this
together with 3.2.2 implies then that E is an injective hull of M . Indeed, assume that E has a proper essential extension E 0 . Since I is injective there exists : E 0 ! I extending the inclusion E I . Suppose Ker = 0 then Im I is an essential extension of M (in I ) properly containing
99
3.2. Injective hulls. Matlis duality
E , a contradiction. On the other hand, since extends the inclusion E I we have E Ker = 0. But this contradicts the essentiality of the extension E E 0 . (b) Since I is injective, can be extended to a homomorphism ' : E I . We have ' M = , and so M Ker ' = Ker = 0. Thus, since the extension M E is essential, we even have Ker ' = 0. (c) By (b) there is a monomorphism ' : E E 0 such that ' M equals the inclusion M E 0 . Im ' is injective and hence a direct summand of E 0 . However, since the extension M E 0 is essential, ' is surjective, and
!
\
j
\
!
j
therefore an isomorphism. We may apply 3.2.4 to construct an injective resolution E (M ) of a module M which for obvious reasons is called the minimal injective resolution of M : we let E 0 (M ) = E (M ), and denote by @;1 the embedding M ! E 0(M ). Suppose the injective resolution has already been constructed up to the i-th step: .
0 ;! E 0(M ) ;! E 1 (M ) ;! ;! E i;1(M ) ;;! E i (M ): We then de ne E i+1(M ) = E (Coker @i;1), and @i is de ned in the obvious way. It is clear that any two minimal injective resolutions of M are isomorphic. Moreover, if I is an arbitrary injective resolution of M , then, as is readily seen, E (M ) is isomorphic to a direct summand of I . We note a technical result about injective hulls which will be needed later in this section. Lemma 3.2.5. Let R be a Noetherian ring, S R a multiplicatively closed set and M an R-module. Then ER (M )S = ERS (MS ): Proof. We show that ER (M )S is an injective hull of the RS -module MS . We know from 3.1.3 that ER (M )S is an injective RS -module. It remains to be shown that ER (M )S is an essential extension of MS . To simplify notation we set N = ER (M ), and pick x 2 NS , x 6= 0. We want to prove that RS x \ MS 6= 0. There exists y 2 N such that RS y = RS x. Thus we may as well assume that x 2 N . We consider the set of ideals S = fAnn(tx): t 2 S g. Since R is Noetherian this set has a maximal element, say Ann(sx), and since RS x = RS (sx), we may replace x by sx, and thus may assume that Ann(x) is maximal in the set S. Since N is an essential extension of M , we have Rx \ M = Ix 6= 0 where I is an ideal in R . Let I = (a1 . . . an ), and assume that ai x = 0 in NS for i = 1 . . . n. Then there exists t 2 S such that t(aix) = 0 in N for i = 1 . . . n. But Ann(tx) = Ann(x), by the choice of x, and so Ix = 0. This is a contradiction. Hence aix 6= 0 in NS for some i, and it follows that RS x \ MS 6= 0. @i;1
@0
.
.
.
100
3. The canonical module. Gorenstein rings
In the next theorem we determine the indecomposable injective R modules of a Noetherian ring R . Recall that an R -module M is decomposable if there exist non-zero submodules M1 M2 of M such that M = M1 M2 otherwise it is indecomposable. Theorem 3.2.6. Let R be a Noetherian ring. (a) For all 2 Spec R the module E (R= ) is indecomposable. (b) Let I 6= 0 be an injective R-module and let 2 Ass I. Then E (R= ) is p
p
p
p
a direct summand of I. In particular, if I is indecomposable, then
I = E (R= ):
(c) Let 2 Spec R. Then E (R= ) = E (R= ) () = . Proof. (a) Suppose E (R= ) is decomposable. Then there exist non-zero submodules N1 N2 of E (R= ) such that N1 \ N2 = 0. It follows that (N1 \ R= ) \ (N2 \ R= ) = (N1 \ N2 ) \ R= = 0. On the other hand, since R= E (R= ) is an essential extension, we have N1 \ R= 6= 0 6= N2 \ R= . This contradicts the fact that R= is a domain. (b) R= may be considered as a submodule of I since 2 Ass I . It follows from 3.2.4 that there exists an injective hull E (R= ) of R= such that E (R= ) I . As E (R= ) is injective, it is a direct summand of I . Statement (c) follows from the next lemma. Lemma 3.2.7. Let R be a Noetherian ring, 2 Spec R, and M a nite p
q
p
p
q
p
q
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
R-module. Then (a) Ass M = Ass E (M ) in particular one has = Ass E (R= ) (b) k( ) = HomR (k( ) E (R= ) ). Proof. (a) It is clear that Ass M Ass E (M ). Conversely, suppose Ass E (M ). Then there exists a submodule U E (M ) which is isomorphic to R= . We have U M = 0 since the extension M E (M ) is essential, and so Ass(U M ) Ass M . (b) Since E (R= ) = ER (k( )), we assume that (R k) is local and = is the maximal ideal. The k-vector space HomR (k E (k)) may be identi ed with V = x E (k): x = 0 it contains k. If V = k, then there exists a non-zero vector subspace W of V with k W = 0. This, however, contradicts the essentiality of the extension k E (k). The importance of the indecomposable injective R -modules results p
q
fg p
p
p
2
q
q
p
p
p
m
\ 6 \
p
p
p
2 f 2
p
m
p
g
m
\
6
from the following: Theorem 3.2.8. Every injective module I over a Noetherian ring R is a
direct sum of indecomposable injective R-modules, and this decomposition is unique in the following sense: for any Spec R the number of indecomposable summands in the decomposition of I which are isomorphic to E (R= ) depends only on I and (and not on the particular decomposition). In fact, this number equals dimk( ) HomR (k( ) I ). p
p
2
p
p
p
p
p
101
3.2. Injective hulls. Matlis duality
Consider the set S of all subsets of the set of indecomposable injective submodules of I with the property: if F 2 S, then the sum of all modules belonging to F is direct. The set S is partially ordered by inclusion. By Zorn's lemma it has a maximal element F0 . Let E be the sum of all the modules in F0. The module E is a direct sum of injective modules, and hence by 3.1.3 is itself injective. Therefore E is a direct summand of I , and we can write I = E H , where H is injective since it is a direct summand of I . Suppose H 6= 0 then there exists 2 Ass H , and so E (R= ) is a direct summand of H see 3.2.6(b). Thus we may enlarge F0 by E (R= ), contradicting the maximality of F0. We conclude that H = 0 and I = EL . Suppose that I = 2 I is the given decomposition. Then Proof.
p
p
p
HomR (k( ) I ) = HomR (k( ) p
M
p
p
p
p
2
By 3.2.7 we have
M
HomR (k( ) (I ) ) =
(I ) ) = p
M
p
2
p
p
M
HomR (k( ) (I ) ): p
2
p
p
HomR (k( ) (I ) ) p
2 0
p
p
where 0 = f 2 : I = E (R= )g. If we again use 3.2.7, we nally get
M p
HomR (k( ) I ) = p
p
p
2 0
HomR (k( ) (I ) ) = k( )( ): 0
p
p
p
p
Bass numbers. Let R be a Noetherian ring, M a nite R -module and Spec R . The ( nite) number i ( M ) = dimk( ) ExtiR (k( ) M ) is called the i-th Bass number of M with respect to .
2
p
p
p
p
p
p
These numbers have an interpretation in terms of the minimal injective resolution of M . Proposition 3.2.9. Let R be a Noetherian ring, M a nite R-module, and E (M ) the minimal injective resolution of M. Then p
.
M
E i (M ) =
p
Let 0 ;! M ;! E 0(M ) injective resolution of M , and let it follows from 3.2.5 that p
0 ;! M
p
p
2Spec R
Proof.
E (R= )i (
M ):
@ @ ;! E 1(M ) ;! be the minimal 2 Spec R. Since localization is exact, 0
1
d d ;! E0(M) ;! E 1(M ) ;! 0
p
p
1
p
is the minimal injective resolution of M here d i is the localization of @i . p
102
3. The canonical module. Gorenstein rings
The complex HomR (k( ) E (M ) ) is isomorphic to the subcomplex C of E (M ) , where .
p
p
p
.
.
p
f 2 Ei(M) :
Ci = x
p
p
g
R x=0 : p
Let x be a non-zero element of C i . Since the extension Im d i;1 E i (M ) is essential, there exists a 2 R with ax 2 Im d i;1 and ax 6= 0. Since R annihilates x, we see that a 2= R . Hence a is a unit in R , and x 2 Im d i;1. It follows that d i (x) = 0, and hence d i jC i = 0 for all i. Consequently we get ExtiR (k( ) M ) = HomR (k( ) E i (M ) ), which by 3.2.8 implies the isomorphism asserted. Among the Bass numbers the type of a module or a local ring is of particular importance. Let (R k) be a Noetherian local ring and M a nite module of depth t. In Chapter 1 we have already considered the Bass number r(M ) = t ( M ), and called it the type of M . In the next theorem we give a new, extremely useful characterization of Gorenstein rings. Theorem 3.2.10. Let (R k) be a Noetherian local ring. The following p
p
p
p
p
p
p
p
p
p
p
p
p
m
m
m
conditions are equivalent: (a) R is a Gorenstein ring (b) R is a Cohen{Macaulay ring of type 1.
Let x be a maximal R -sequence. By 3.1.19, R is Gorenstein if and only if R=(x) is. Similarly the properties in (b) are stable under specialization modulo x see 1.2.19 and 2.1.3. Thus we may assume dim R = 0. (a) ) (b): By 3.1.17, R is an injective R -module. Since R is local, it is indecomposable as an R -module, and so, since Ass R = f g, we have that R = ER (k) see 3.2.6. It follows from 3.2.7 that R is of type 1. (b) ) (a) follows from statement (e) in 3.2.12 below. We use this new characterization of Gorenstein rings to give examples of Cohen{Macaulay rings which are not Gorenstein, and of Gorenstein rings which are not complete intersections. Examples 3.2.11. (a) Let (R k) be an Artinian local ring for which 2 = 0. For instance, R = kX . . . X ]=(X . . . X )2 is such a ring. It 1 n 1 n is easily seen that = Soc R . Hence we have r(R ) = emb dim R , and conclude that R is Gorenstein if and only if emb dim R = 1. When R is Gorenstein, it is even a complete intersection. (b) In the following we present a method to produce a large class of Artinian Gorenstein rings: let k be a eld, S = kX1 . . . Xn] the polynomial ring in n variables over k, m an integer, Sm the m-th homogeneous part of S , and ' : Sm ! k a non-trivial k-linear map. Proof.
m
m
m
m
103
3.2. Injective hulls. Matlis duality
For every jL2 N, we de ne Ij = fa 2 Sj : '(a Sm;j ) = 0g. It is readily seen that I = j 0 Ij is a graded ideal with Ij = Sj for j > m. Thus we conclude that R = S=I is an Artinian (graded) local ring. We claim that R is a Gorenstein ring. To see this, we determine the socle of R . For any element a 2 S , we denote by a its residue class modulo I . Let j 2 N with 0 < j < m, and let a 2 Rj a 6= 0. Then by the de nition of I , there exists b 2 Sm;j such that '(a b) 6= 0, and so a b 6= 0. But since b belongs to the maximal ideal of R , it follows that a 2= Soc R . Therefore, Soc R = Rm . As dimk Rm = 1, it follows that R is Gorenstein. We give an explicit example for this construction: let ' : S2 ! k be the k-linear map with '(XiXj ) = 0 1
i < j n
'(Xi2) = 1 i = 1 . . . n:
For this linear form ' we get I = (X12
; X22 . . . X12 ; Xn2 X1X2 X1X3 . . . Xn;1Xn):
Therefore, R = S=I is Gorenstein, and is a complete intersection if and only if n 2. Matlis duality. Let (R k) be a Noetherian local ring. We are going to study the functor which takes the dual M 0 of an R -module M with respect to the injective hull E of k. If M is a nite module, the dual M 0 need not be nite. Indeed, we know from Exercise 3.1.23 that R 0 = E is nite only if R is Artinian. However, the E -dual of a module of nite m
length also has nite length, as we shall see now. Proposition 3.2.12. Let (R k) be a Noetherian local ring, E the injective m
hull of k, and N an R-module of nite length. For any R-module M we set M 0 = HomR (M E ). Then: (a) one has i = 0, ExtiR (k E ) = k0 for for i > 0
(b) `(N ) = `(N 0 ) (c) the canonical homomorphism N ! N 00 is an isomorphism (d) (N ) = r(N 0 ) and r(N ) = (N 0 ) (e) if R is Artinian, then E is a nite faithful R-module satisfying (i) `(E ) = `(R ), (ii) the canonical homomorphism R ! EndR (E ), a 7! 'a, where 'a(x) =
2
ax for all x E, is an isomorphism, (iii) r(E ) = 1 and (E ) = r(R ) conversely, any nite faithful R-module of type 1 is isomorphic to E.
104
3. The canonical module. Gorenstein rings
(a) ExtiR (k E ) = 0 for i > 0, as E is injective furthermore HomR (k E ) = k see 3.2.7. (b) We prove the equality asserted by induction on the length of N . If `(N ) = 1, then N = k, and the equality follows from (a). Now suppose that `(N ) > 1. Then there exists a proper submodule U N , and we obtain an exact sequence 0 ;! U ;! N ;! W ;! 0 with `(U ) < `(N ) and `(W ) < `(N ). Since E is injective this sequence yields the dual exact sequence 0 ;! W 0 ;! N 0 ;! U 0 ;! 0: The induction hypothesis applies to U and W , and the additivity of length gives the result. (c) Again we use induction on `(N ). If `(N ) = 1, then N = k, and N 00 = k by (a). Therefore it suces to show that the canonical homomorphism : k ! HomR (HomR (k E ) E ) is not the zero map. Let x 2 E , x 6= 0, be a socle element of E . There exists ' 2 HomR (k E ) with '(1) = x. Then (1)(') = x 6= 0, and so 6= 0. If `(N ) > 1, we choose as before an exact sequence 0 ;! U ;! N ;! W ;! 0 with `(U ) < `(N ) and `(W ) < `(N ). The natural homomorphisms into the bidual modules induce a commutative diagram 0 ;;;;! U ;;;;! N ;;;;! W ;;;;! 0 Proof.
?? y
?? y
?? y
0 ;;;;! U 00 ;;;;! N 00 ;;;;! W 00 ;;;;! 0 where the outer vertical arrows are isomorphisms by our induction hypothesis. The snake lemma (318], Theorem 6.5) applied to this diagram implies N ! N 00 is an isomorphism. (d) The module (N= N )0 is the kernel of the linear map N 0 ! ( N )0 which assigns to every ' 2 N 0 its restriction to N . Hence ' 2 (N= N )0 if and only if '(N ) = '( N ) = 0. In other words, (N= N )0 = f' 2 N 0 : ' = 0g = Soc N 0 : Thus we get (N ) = dimk N= N = dimk (N= N )0 = dimk Soc N 0 = r(N 0 ). The second equality follows from the rst by (c). (e) By (b) we have `(E ) = `(R 0) = `(R ) < 1. In particular, E is a nite R -module. Next it follows from (c) that the canonical homomorphism : R ! HomR (HomR (R E ) E ) is an isomorphism. If we identify m
m
m
m
m
m
m
m
m
m
105
3.2. Injective hulls. Matlis duality
HomR (R E ) with E , then identi es with the canonical homomorphism R ! EndR (E ). A module whose endomorphism ring is R is necessarily faithful. Statement (e)(iii) follows from (d). Finally, let N be a faithful R -module of type 1. Then N 0 is cyclic, and so N = HomR (R=I E ) for some ideal I . Here we have used (c) and (d). But since N is faithful, I = 0 and so N = E. Proposition 3.2.12 may be viewed as the Matlis duality theorem for nite Artinian modules. Now we prove its general form. It will be of crucial importance for the local duality theorem of Grothendieck, which we will discuss in Section 3.5. Let (R k) be a complete local ring. We denote by M(R ) the category of R -modules, by A(R ) the full subcategory of Artinian R modules and by F(R ) the full subcategory of nite R -modules. Let E be an injective hull of k. We set T ( ) = HomR ( E ). The contravariant functor T : M(R ) ! M(R ) is exact. Its restriction to A(R ) or F(R ) will again be denoted by T . m
Theorem 3.2.13 (Matlis). Let (R k) be a Noetherian complete local ring, N 2 A(R ) and M 2 F(R ). Then (a) T (R ) = E and T (E ) = R, (b) T (M ) 2 A(R ) and T (N ) 2 F(R ), (c) there are natural isomorphisms T (T (N )) = M, = N and T (T (M )) (d) the functor T establishes an anti-equivalence between the categories A(R) and F(R). m
We proceed in several stages. (1) For all n 2 N we set En = n x = 0g. Let x 2 E , x 6= 0 then Ass(Rx) Ass E = f g, see 3.2.7.SHence there exists an integer n such that nx = 0. This proves that E = n 0 En = lim ;! En. (2) The natural homomorphism R ! EndR (E ) = T (E ) is an isomorphism: by 3.2.12(e)(ii), the natural homomorphisms n : R= n ! T (En) are isomorphisms, and we obtain commutative diagrams such as Proof.
fx 2 E :
m
m
m
m
?? ;;;;! T ??(E) y y
R R=
m
n
;;;;! T (En) n
in which the only homomorphisms are the natural ones. As R is complete, the map R ! lim R= n is an isomorphism. Likewise T (E ) ! lim ; ; T (En) is an isomorphism since by 318], Theorem 2.27, and (1) we have lim = T (E ). It follows that the natural homomorphism ; T (ERn)!=TT(E(lim ;)!isEann) isomorphism as well. This proves (a). m
106
3. The canonical module. Gorenstein rings
(3) E is Artinian: let E = U0 U1 U2 be a descending chain of submodules of E . This chain induces a sequence of epimorphisms
;! T (U1) ;! T (U2) ;! Thus we can write T (Ui) = R=Ii, where (0) = I0 I1 I2 is an ascending chain of ideals. Since R is Noetherian this chain stabilizes, and so there exists an integer i0 such that T (Ui) = T (Ui+1) for i i0 . We will show that Ui = Ui+1 for i i0 . Suppose that Ui = 6 Ui+1, but T (Ui) = T (Ui+1). Let V = Ui =Ui+1 then V = 6 0, but T (V ) = 0. However, V is a subquotient of E , and so Ass V = f g see 3.2.7. In other words, there exists a monomorphism k ! V . Applying T , we obtain an epimorphism 0 = T (V ) ! T (k) = k, a contradiction. (4) If N is Artinian, then there exists an embedding N ! E n for some integer n: Soc N is a nite dimensional k-vector space since N is Artinian. Moreover, the extension Soc N N is essential. In fact, if x 2 N , then Rx is a nite Artinian module, and therefore Rx \ Soc N = Soc Rx = 6 0. Let N I be an embedding of N into an injective R -module. By 3.2.4, an injective hull E (N ) can be chosen as a maximal essential extension of N in I . Since the extension Soc N N is essential, E (N ) is likewise an injective hull of Soc N . Suppose Soc N = kn then it follows that n N E (Soc N ) = E . The remaining assertions of the theorem now follow easily. (b) Let N 2 A(R ) then by (4) there exists an embedding N E n which by (2) induces an epimorphism R n ! T (N ) therefore T (N ) 2 F(R). Conversely, suppose M 2 F(R). We choose an epimorphism R n ! M . This epimorphism yields an embedding T (M ) ! E n . The module E is Artinian by (3), and so any submodule of E n is Artinian. It follows that T (M ) 2 A(R ). (c) By (4) there exists an integer n and an exact sequence 0 ! N ! E n ! W ! 0 which we may complete to a commutative diagram n 0 ;;;;! N ? ;;;;! E? ;;;;! W? ;;;;! 0 R = T (E )
m
?y
?y
?y
0 ;;;;! T (T (N )) ;;;;! T (T (E n)) ;;;;! T (T (W )) ;;;;! 0 whose vertical maps are just the canonical homomorphisms. It follows from (2) that is an isomorphism. Therefore, by the snake lemma, is an isomorphism if and only if is a monomorphism. Let x 2 Ker then '(x) = 0 for all ' 2 HomR (W E ). Suppose x 6= 0, and let : Rx ! E be the homomorphism which maps x to a non-zero socle element of E . Then (x) 6= 0, and since E is injective, can be extended to a homomorphism ' : W ! E . We then have '(x) 6= 0, a contradiction.
3.3. The canonical module
107
Similarly one proves that the natural linear map M ! T (T (M )) is an isomorphism, starting with the exact sequence 0 ! U ! R n ! M ! 0 and using the fact that the natural homomorphism R ! T (T (R )) is an isomorphism (which is an immediate consequence of (2)). Exercises Let (R k) be a Noetherian local ring, E an injective hull of k. Prove: (a) The natural homomorphism E ! E R R^ is an isomorphism. In particular, E is an R^ -module. (b) As an R^ -module, E
= ER^ (k). (c) For all nite R -modules N there exists a natural isomorphism HomR (N E )
= HomR^ (N^ E ): (If this problem seems to be too di cult, the reader may consult 270], Theorem 18.6.) 3.2.15. Let (R ) be an Artinian local ring. Show the following conditions are equivalent: (a) R is a Gorenstein ring (b) all nite R -modules are re exive (c) I = Ann Ann I for all ideals I of R (d) for all non-zero ideals I and J one has I \ J 6= 0. 3.2.14.
m
m
3.3 The canonical module So far we have studied nite modules of nite injective dimension over Noetherian local rings, but we have ignored the question as to under what circumstances such modules actually exist. A Gorenstein ring R admits plenty of nite modules of nite injective dimension: any module of nite projective dimension has nite injective dimension as well, simply because R itself has nite injective dimension by de nition. Also any Artinian local ring (R k), Gorenstein or not, admits a nite injective module { the injective hull of k. The question becomes more delicate for non-Gorenstein local rings of positive dimension. One of the main results of this section will be that any Cohen{Macaulay ring which is a homomorphic image of a Gorenstein ring has a nite module of nite injective dimension. Moreover, this module can be chosen to be a maximal Cohen{Macaulay module of type 1. It will be shown that such a module is unique up to isomorphism. It is called the canonical module of R . For a Gorenstein ring the canonical module is just the ring itself. We shall study the behaviour of the canonical module under at extensions, localizations, and specializations. De nition 3.3.1. Let (R k) be a Cohen{Macaulay local ring. A maximal Cohen{Macaulay module C of type 1 and of nite injective dimension is called a canonical module of R . m
m
108
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It is immediate (see 1.2.8 and 3.1.14) that C is a canonical module of
R if and only if
dimk ExtiR (k C ) = id d = dim R: Two questions arise: when does a canonical module exist, and is it uniquely determined up to isomorphism? This question has a simple answer in the case dim R = 0: by 3.2.8, ER (k) is the uniquely determined canonical module. To prove uniqueness in general, we will need the following two results. Lemma 3.3.2. Let (R k) be a Noetherian local ring, ' : M ! N a homomorphism of nite R-modules, and x an N-sequence. If ' R=(x) is m
an isomorphism, then ' is an isomorphism. Proof. The surjectivity of ' follows from Nakayama's lemma. In order to prove that ' is injective, we may assume without loss of generality that the sequence x consists of one element, say x. Let K = Ker ' since x is N -regular, the exact sequence 0 K M N 0
;! ;! ;! ;!
induces the exact sequence 0 ;! K=xK ;! M=xM ;! N=xN ;! 0: By assumption, K=xK = 0, and hence K = 0, by Nakayama's lemma. Proposition 3.3.3. Let (R k) be a Cohen{Macaulay local ring of dimenm
sion d, and C a maximal Cohen{Macaulay R-module. (a) Suppose M is a maximal Cohen{Macaulay R-module with ExtjR (M C ) = 0 for all j > 0. Then HomR (M C ) is a maximal Cohen{Macaulay module, and for any R-sequence x we have HomR (M C ) R=xR = HomR=xR (M=xM C=xC ): (b) Assume in addition that C has nite injective dimension, and M is a Cohen{Macaulay R-module of dimension t. Then (i) ExtjR (M C ) = 0 for j = d t, (ii) ExtdR;t(M C ) is a Cohen{Macaulay module of dimension t. Proof. (a) Let x be an R -regular element. Since C is a maximal Cohen{Macaulay module, the element x is C -regular as well, and one
6 ;
2
m
has the exact sequence
0 ;! C ;! C ;! C=xC ;! 0 which by our assumtion induces the exact sequence x
0 ;! HomR (M C ) ;! HomR (M C ) ;! HomR (M C=xC ) ;! 0: x
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3.3. The canonical module
Therefore,
HomR=xR (M=xM C=xC ) = HomR (M C=xC ) = HomR (M C )=x HomR (M C ) = HomR (M C ) R=xR:
For an arbitrary R -sequence x one proceeds by induction on the length of the sequence. (b)(i) It follows from 1.2.10(e) that ExtjR (M C ) = 0 for j < d ; t. Next we show by induction on t that ExtjR (M C ) = 0 for any t-dimensional Cohen{Macaulay module M and all j > d ; t. If t = 0, then the claim follows from 3.1.17. Now suppose that t > 0, and let x 2 be an M -regular element. The exact sequence m
0 ;! M ;! M ;! M=xM ;! 0 x
induces the exact sequence
ExtjR (M C ) ;! ExtjR (M C ) ;! ExtjR+1(M=xM C ): x
; 1)-dimensional Cohen{Macaulay module. Hence by the induction hypothesis we have ExtjR+1(M=xM C ) = 0 for j > d ; t, and so Nakayama's lemma implies that ExtjR (M C ) = 0 for j > d ; t. (ii) We proceed by induction on t. The assertion is trivial if t = 0. Assume now dim M = t > 0, and let x 2 be an M -regular element. By (i), the exact sequence
M=xM is a (t
m
0 ;! M ;! M ;! M=xM ;! 0 x
yields the exact sequence
0 ;! ExtdR;t(M C ) ;! ExtdR;t(M C ) ;! ExtdR;(t;1)(M=xM C ) ;! 0: x
Thus x is regular on ExtdR;t(M C ), and so it follows from our induction hypothesis that ExtdR;t(M C ) is Cohen{Macaulay. We are now ready to prove the uniqueness of the canonical module. Theorem 3.3.4. Let (R k) be a Cohen{Macaulay local ring, and let C and C 0 be canonical modules of R. Then (a) C=xC = ER=(x)(k) for any maximal R-sequence x, (b) the canonical modules C and C 0 are isomorphic, (c) HomR (C C 0) = R, and any generator ' of Hom(C C 0) is an isomorphism, (d) the canonical homomorphism R EndR (C ) is an isomorphism. m
!
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3. The canonical module. Gorenstein rings
(a) By 3.1.15, C=xC is an injective R=(x)-module of type 1. Since Spec(R=(x)) = f =(x)g, 3.2.8 yields the assertion. (b) and (c): It follows from (a) that C=xC = ER=(x) (k) = C 0=xC 0: Now 3.3.3 and 3.2.12 imply that HomR (C C 0) R R=(x) = HomR=(x) (C=xC C 0=xC 0) = R=(x) 0 and so HomR (C C ) is cyclic by Nakayama's lemma. Let ' be a generator of this module. Then the natural inclusion R' ! HomR (C C 0) induces the above isomorphism modulo x. By 3.3.3, HomR (C C 0) is a maximal Cohen{Macaulay module. Thus 3.3.2 implies that R' ! HomR (C C 0) is an isomorphism. In particular it follows that R' is a maximal Cohen{ Macaulay module. We may therefore apply 3.3.2 once again to conclude that R ! R' is an isomorphism, too. Next we show that ' : C ! C 0 is an isomorphism. Indeed, ' R=(x) can be identi ed with a generator of End(ER=(x)(k)). It follows therefore from 3.2.12(e)(ii) that ' R=(x) is an isomorphism. Since C 0 is a maximal Cohen{Macaulay module, 3.3.2 implies that ' is an isomorphism. It is clear that any other isomorphism C ! C 0 is a generator of HomR (C C 0), too. (d) is proved similarly. In view of this result we may talk of the canonical module of R provided it exists. From now on we will denote the canonical module of R by !R . The next theorem lists some useful and often applied change of ring formulas for the canonical module. Theorem 3.3.5. Let (R k) be a Cohen{Macaulay local ring with canonProof.
m
m
ical module !R . Then (a) !R =x!R = !R=xR for all R-sequences x, that is, the canonical module specializes, (b) (!R ) = !R for all Spec R, that is, the canonical module localizes, (c) (!R )b = !R^ . Proof. (a) First notice that x is an !R -sequence, too. The (R=xR )-module !R =x!R has nite injective dimension see 3.1.15. Since r(!R =x!R ) = r(!R ) = 1, the module !R =x!R is the canonical module of R=xR , by p
p
p
2
de nition. (b) The R -module (!R ) has nite injective dimension (see 3.1.9), and is again a maximal Cohen{Macaulay module. It remains to be shown that r((!R ) ) = 1. Let x be a sequence of elements of R whose image in R is a maximal R -sequence. Then by 3.1.15, M = (!R ) =x(!R ) p
p
p
p
p
p
p
111
3.3. The canonical module
is an injective module over the Artinian local ring A = R =xR . It follows from 3.2.7 and 3.2.8 that M r = r (M ): = EA (k( ))r From 3.2.12 we get p
p
p
(1) HomA (M M ) = Ar : On the other hand, from 3.3.5(a) we obtain (2) HomA (M M ) = HomR=xR (!R =x!R !R =x!R ) = HomR=xR (!R=xR !R=xR ) = (R=xR) = A: For the last isomorphism we used the fact that the endomorphism ring of the canonical module of S = R=xR is isomorphic to S see 3.3.4. A comparison of (1) and (2) yields r = 1, as desired. (c) The bre of R ! R^ is k, so that by atness, ExtiR (k !R )b = ExtiR^ (k (!R )b) for all i. This implies the assertion. 2
p
p
p
Existence of the canonical module. Our next goal is to clarify for which
Cohen{Macaulay local rings the canonical module exists. Theorem 3.3.6. Let (R k) be a Cohen{Macaulay local ring. The folm
lowing conditions are equivalent: (a) R admits a canonical module (b) R is the homomorphic image of a Gorenstein local ring.
One direction of the proof resorts to the principle of idealization due to Nagata: let R be a ring and M an R -module. We construct a ring extension R R M of R , called the trivial extension of R by M . As an R -module, R M is just the direct sum of R and M . The multiplication is de ned by (a x)(b y) = (ab ay + bx) for all a b 2 R and x y 2 M . Some basic facts on trivial extensions are the subject of Exercise 3.3.22. Here we will only use that R M is a ring, and if M is nite and R is a Noetherian (or Artinian) local ring with maximal ideal , then so is R M with maximal ideal M = f(a x) 2 R M : a 2 g. Proof of 3.3.6. (a) ) (b): The ring R is a homomorphic image of the trivial extension R !R . We will show that R !R is a Gorenstein ring. Let x be an R -regular sequence of maximal length. It is easy to see that x is a maximal (R !R )-sequence as well, and that (R !R )=x(R !R ) = (R=xR ) (!R =x!R ): m
m
m
112
3. The canonical module. Gorenstein rings
By 3.3.4, !R =x!R = ER=xR (k). Bearing in mind the characterization 3.2.10 of Gorenstein rings, we may assume that R is Artinian, and it remains to be shown that the type of the Artinian local ring R 0 = R ER (k) is 1. Let (a x) 2 Soc R 0 then (b 0)(a x) = (ba bx) = (0 0) for all b 2 . This implies that a 2 Soc R and x 2 Soc ER (k). Assume that a 6= 0. The exact sequence a R ;! R ;! R=(a) ;! 0 a induces the exact sequence 0 ;! ER=(a)(k) ;! ER (k) ;! ER (k) (see 3.1.6). As `(ER=(a)(k)) = `(R=(a)) < `(R ) = `(ER (k)) (see 3.2.12), multiplication by a on ER (k) cannot be the zero map. Therefore there exists y 2 ER (k) with ay 6= 0, and so (0 y)(a x) = (0 ay) 6= (0 0), a contradiction. Our conclusion is that Soc(R ER (k)) = Soc ER (k), and therefore by 3.2.12, r(R E (k)) = 1. For the proof of 3.3.6(b) ) (a) we note the following more general result. Theorem 3.3.7. Let (R ) be a Cohen{Macaulay local ring. (a) The following conditions are equivalent: (i) R is Gorenstein (ii) !R exists and is isomorphic to R. (b) Let ' : (R ) ! (S ) be a local homomorphism of Cohen{Macaulay m
m
m
n
local rings such that S is a nite R-module. If !R exists, then !S exists and !S = ExttR (S !R ) t = dim R dim S:
;
(a)(i) () (ii) follows from 3.1.18 and 3.2.10. (b) By virtue of 2.1.2(b), and since dim S = dim(R= Ker '), there exists an R -sequence x = x1 . . . xt with xi 2 Ker ', t = dim R ; dim S . Set R = R=(x)R as !R =(x)!R = !R (see 3.3.5), we have ExttR (S !R ) = HomR (S !R ), by 3.1.16. Thus we may assume from the beginning that dim R = dim S . Let d = dim R , and y = y1 . . . yd an R -sequence. Then y is !R regular and HomR (S !R )-regular as well, since both modules are Cohen{ Macaulay modules of dimension d see 3.3.3. It follows from 3.3.3(a) that HomR (S !R ) R R 0 = HomR0 (S 0 !R0 ) where R 0 = R=(y )R , and S 0 = S=(y )S . In view of Exercise 3.3.23 it suces to show that HomR0 (S 0 !R0 ) is the canonical module of S 0 . Since !R0 = ER0 (k), 3.1.6 implies that HomR0 (S 0 ER0 (k)) is an injective S 0 -module, and so HomR0 (S 0 ER0 (k)) = ES 0 (k)r for some r > 0. By 3.2.12(b) and (e)(i) Proof.
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3.3. The canonical module
we get `(ES 0 (k)) = `(S 0 ) = `(HomR0 (S 0 ER0 (k))) = r `(ES 0 (k)) therefore r = 1. A noteworthy case of 3.3.7 is the following: let k be a eld, and R an Artinian local k-algebra. Then Homk (R k) is the canonical module of R . A Noetherian complete local ring is a homomorphic image of a regular local ring see A.21. Regular local rings are Gorenstein (see 3.1.20), and so 3.3.6 implies Corollary 3.3.8. A complete Cohen{Macaulay local ring admits a canonical module.
Corollary 3.3.9. Let (R k) be a regular local ring and I an ideal of height g such that S = R=I is Cohen{Macaulay. Let F : 0 ;! Fg ;! Fg;1 ;! ;! F0 ;! 0 be the minimal free R-resolution of S, and let G = HomR (F R ) be the m
m
.
dual complex
.
;! ;!
.
;! ;! ;!
G. : 0 Gg Gg;1 G0 0 where Gi = Fg;i for i = 0 . . . g. Then G. is a minimal free R-resolution of !S .
Note that g is indeed the length of the minimal free resolution of S see 2.2.10. One has ExtiR (S R ) = H i (F. ) for all i 0. The corollary Proof.
follows therefore from 3.3.3(b) and 3.3.7.
Further properties of the canonical module. In the next theorem some
useful characterizations of the canonical module will be given. Theorem 3.3.10. Let (R k) be a Cohen{Macaulay local ring of dimenm
sion d, and let C be a nite R-module. Then the following conditions are equivalent: (a) C is the canonical module of R (b) i ( C ) = ih for all i 0 and all Spec R, where h = height (c) for all integers t = 0 1 . . . d, and all Cohen{Macaulay R-modules M of dimension t one has (i) ExtRd ;t(M C ) is a Cohen{Macaulay R-module of dimension t, (ii) ExtiR (M C ) = 0 for all i = d t, (iii) there exists an isomorphism M ExtdR;t (ExtdR;t(M C ) C ) which in the case d = t is just the natural homomorphism from M into the bidual of M with respect to C (d) for all maximal Cohen{Macaulay R-modules M one has (i) HomR (M C ) is a maximal Cohen{Macaulay R-module, (ii) ExtiR (M C ) = 0 for i > 0, (iii) the natural homomorphism M HomR (HomR (M C ) C ) is an isomorphism. p
p
6 ;
2
!
!
p
114
3. The canonical module. Gorenstein rings
(a) () (b): The canonical module localizes see 3.3.5 therefore (a) implies (b). Choosing = , we obtain (a) from (b). (a) ) (c): (i) and (ii) have already been shown in 3.3.3. From the Rees lemma 3.1.16 and (i) one deduces that Proof.
p
m
ExtdR;t (ExtdR;t(M C ) C ) = HomR=xR (HomR=xR (M C=xC ) C=xC )
for an R -sequence x of length d ; t which is contained in AnnR M . Replacing R by R=xR , we may as well assume that t = d . Since by 3.3.3, HomR (M C ) R=y R = HomR=yR (M=y M C=y C ) for any R -sequence y , we may nally assume that dim R = 0. In this case however C = ER (k), and the assertion follows from 3.2.12. (d) is a special case of (c). (d) ) (a): If we choose M = R , then it follows from (i) that C is a maximal Cohen{Macaulay module. According to Exercise 2.1.26, for all i d the i-th syzygy module of the residue class eld k of R is 0 or a maximal Cohen{Macaulay module. Therefore (ii) implies that ExtiR (k C ) = 0 for i > d , and hence we have inj dim C < 1 see 3.1.14. It remains to be shown that r(C ) = 1. By 3.3.3(a), the conditions in (d) are stable under reduction modulo R -sequences. Thus, since the type of C is also stable under reduction modulo R -sequences, we may restrict ourselves to the case where R is Artinian. Then the module C is necessarily injective, and so it must be isomorphic to ER (k)r , r = r(C ). Now it follows from 3.2.12 that R r = HomR (HomR (R C ) C ). Consequently condition (iii) implies r(C ) = 1. We complement the previous theorem with some extra information about the Exti ( !R ). Observe the analogy of the statements with 3.2.12. The canonical module takes the position of the injective hull when one deals with arbitrary Cohen{Macaulay local rings rather than Artinian local rings. Proposition 3.3.11. Let R be a Cohen{Macaulay local ring of dimension 2
d with canonical module !R , and M a Cohen{Macaulay R-module of dimension t. Then (a) (M ) = r(ExtdR;t (M !R )), (b) r(M ) = (ExtdR;t(M !R )), (c) !R is a faithful R-module, and (i) r(!R ) = 1, (!R ) = r(R ), (ii) End(!R ) = R. Proof.
get
There exists an !R -sequence x of length d ; t in Ann M , and we ExtRd ;t (M !R ) = HomR=xR (M !R=xR ):
115
3.3. The canonical module
So we may assume that dim R = dim M . By 3.3.3, we may further assume that dim R = 0. Since the canonical module of an Artinian local ring is the injective hull of the residue class eld, all assertions follow from 3.2.12. The previous proposition has an interesting application. Corollary 3.3.12. Let R be a Cohen{Macaulay local ring, M a Cohen{ Macaulay R-module and 2 Supp M. Then r(M ) r(M ). ^ R^ ) then dim R= ^ = dim R= see 2.1.15. ThereProof. Pick 2 Ass(R= fore we obtain a at local homomorphism R ! R^ whose bre is of dimension zero. From 1.2.16 it follows that r(M ) r(M^ ). Since R ! R^ is at with bre k, 1.2.16 once again applied gives r(M ) = r(M^ ). We may therefore assume that R is complete. By A.11, R is the epimorphic image of complete regular local S , and by Exercise 1.2.26(c) we have rS (M ) = rR (M ). Thus we may assume that R is regular. In particular R is Gorenstein. Hence, by 3.3.7, R has a canonical module and is isomorphic to R , and so 3.3.11 yields r(M ) = (ExtdR;t(M R )) = (ExtdR;t(M R ) ) (ExtdR;t(M R)) = r(M) where d = dim R and t = dim M . Here we have used that, by 2.1.3, d ; t = dim R ; dim M = dim R ; dim R= ; (dim M ; dim M= M ) = dim R ; dim M : p
p
q
p
q
p
p
q
p
p
p
p
q
p
p
p
p
p
p
The canonical module and at extensions. We will show that the canonical module behaves well under at ring extensions. For the proof we need Proposition 3.3.13. Let (R ) be a Cohen{Macaulay local ring, and C a m
nite R-module. The following conditions are equivalent: (a) C is the canonical module of R (b) C is a faithful maximal Cohen{Macaulay R-module of type 1.
Proof. (a) ) (b): The canonical module is a maximal Cohen{Macaulay module of type 1, by de nition, and faithful by 3.3.11. (b) ) (a): Note rst that C has one of the properties in (a) or (b) if and only if the completion C^ has this property. For instance, the property of being faithful means that the canonical homomorphism ' : R ! HomR (C C ) is injective. Since R ! R^ is faithfully at, ' is injective if and only if its completion is injective. The reader should check the other properties. We may now assume that R is complete. Then, by 3.3.8, R has a canonical module !R . By 3.3.11(b), HomR (C !R ) is a cyclic module, say
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R=I , so that by 3.3.10(d)(iii) we have C = HomR (R=I !R ). It follows that I annihilates C . Since we assume that C is faithful, we get I = 0, and hence C = HomR (R !R ) = !R .
Theorem 3.3.14. Let (R ) be a Cohen{Macaulay local ring, and (R ) ! (S ) a at homomorphism of local rings. (a) If !R exists and S= S is Gorenstein, then !S = !R S. (b) If C is a nite R-module, and S a Cohen{Macaulay ring with canonical module !S = C S, then S= S is Gorenstein and C = !R . Proof. A at local homomorphism is faithfully at. Thus we see as in the proof of the previous proposition that a nite R -module M is faithful if and only if M R S is a faithful S -module. By 1.2.16, we have r(!R S ) = r(S= S )r(!R ) = 1 m
m
n
m
m
m
hence (a) follows from 3.3.13. Part (b) is proved similarly: by 1.2.16, 1 = r(S= S )r(C ) and C is a maximal Cohen{Macaulay module. It follows that r(S= S ) = 1 and r(C ) = 1. Therefore S= S is Gorenstein (see 3.2.10), and in view of 3.3.13, C is the canonical module of R . Corollary 3.3.15. Let ' : (R ) ! (S ) be a at homomorphism of m
m
m
m
n
Noetherian local rings. Then S is Gorenstein if and only if R and S= S are Gorenstein. m
The canonical module for non-local rings. We saw in 3.3.5 that the canon-
ical module localizes. This suggests the following De nition 3.3.16. Let R be a Cohen{Macaulay ring. A nite R -module !R is a canonical module of R if (!R ) is a canonical module of R for all maximal ideals of R . Remark 3.3.17. In contrast to the local case, a canonical module is in general not unique (up to isomorphism). Indeed, let R be a Cohen{ Macaulay ring (not necessarily local), and let !R and !R0 be canonical modules of R . We set I = HomR (!R !R0 ). Localizing at a prime ideal and using 3.3.4 and 3.3.5 we see that I = R for all 2 Spec R . We de ne an R -module homomorphism : I !R ! !R0 by (' x) = '(x) for all ' 2 I and all x 2 !R . Then is an isomorphism since it is locally an isomorphism. Conversely, suppose !R is a canonical module of R and I is a locally free R -module of rank 1. Then I !R is locally isomorphic to !R , and so is a canonical module of R . Thus a canonical module of a Cohen{ Macaulay ring is only unique up to a tensor product with a locally free module of rank 1. m
m
m
p
p
p
117
3.3. The canonical module
Proposition 3.3.18. Let R be a Cohen{Macaulay ring and !R a canonical module of R. (a) The following conditions are equivalent: (i) !R has a rank (ii) rank !R = 1 (iii) R is generically Gorenstein, i.e. R is Gorenstein for all minimal prime ideals of R. (b) If the equivalent conditions of (a) hold, then !R can be identi ed with an ideal in R. For any such identi cation, !R is an ideal of height 1 or equals R. In the rst case, the ring R=!R is Gorenstein. Proof. (a)(i) (ii): Let Spec R be a minimal prime ideal. Then !R = (!R ) is a free R -module, and R is Artinian. The canonical module of an Artinian local ring is the injective hull E of the residue class eld, and so the free R -module (!R ) has rank 1 since E is indecomposable see p
p
)
p
p
2
p
p
p
3.2.6. The implications (ii) ) (iii) and (iii) ) (i) are clear in view of 3.3.7. (b) The canonical module !R is torsion-free since all R -regular elements are !R -regular as well. According to Exercise 1.4.18, !R is isomorphic to a submodule of R . Therefore it may be identi ed with an ideal in R which we again denote by !R . If dim R = 0, then necessarily !R = R . We may therefore assume that dim R > 0, and that !R is a proper ideal of R . Then !R must contain an R -regular element since rank !R = 1. Let be a prime ideal containing !R . Using the fact that !R R is a maximal Cohen{Macaulay R -module we then get dim R ; 1 dim(R =!R R ) depth(R =!R R ) depth R ; 1 = dim R ; 1. This shows that height !R = 1, and that R=!R is Cohen{Macaulay. Finally we prove that R=!R is Gorenstein. To show this, we may assume that R is local. Applying the functor HomR ( !R ) to the exact sequence 0 ;! !R ;! R ;! R=!R ;! 0 and using 3.3.4(d) we obtain the exact sequence 0 ;! !R ;! R ;! Ext1R (R=!R !R ) ;! Ext1R (R !R ) = 0: This implies R=!R = Ext1R (R=!R !R ). Thus the conclusion follows from 3.3.7. Corollary 3.3.19. Let R be a Cohen{Macaulay normal domain with canonp
p
p
p
p
p
p
p
p
p
p
p
ical module !R . Then !R is isomorphic to a divisorial ideal. In particular, if R is factorial, then R is Gorenstein. Proof. By 3.3.18, !R is an ideal. It satis es the Serre condition (S2 ), and moreover, (!R ) = !R = R for all prime ideals of height 1. This follows from 3.3.7 since, by normality, R is regular for all prime ideals of height 1. Thus we have shown that !R is a reexive ideal see 1.4.1. p
p
p
p
118
3. The canonical module. Gorenstein rings
A reexive ideal is divisorial. (We refer to Fossum 108] for the theory of divisorial ideals.) In a factorial ring all divisorial ideals are principal, and so !R is principal and R is Gorenstein see 3.3.7. In concluding these considerations we show that formula 3.3.14(a) for the canonical module under at extensions has a non-local counterpart. Proposition 3.3.20. Let ' : R ! S be a at homomorphism of Noetherian rings whose bres S R k( ) are Gorenstein for all 2 Spec R for which there exists a maximal ideal in S with = \ R. If !R is a canonical p
p
R S is a canonical module of S. Proof. Let be a maximal ideal of S , = \ R : then R ! S is a at local homomorphism whose bre is a localization of S R k( ), and thus is Gorenstein. It follows from 3.3.14(a) that !R R S is a canonical module of S . Since (!R R S ) = !R R S , the proposition q
module of R, then !R
p
q
q
p
q
p
q
p
p
q
p
is proved. Corollary 3.3.21. Let R be a Cohen{Macaulay ring with canonical module !R , and let S be either the polynomial ring R X1 . . . Xn] or the formal power series ring R X1 . . . Xn]]. Then !R R S is a canonical module of q
q
p
p
q
S. In particular, if R is Gorenstein, then so is S.
We may assume that n = 1. The result then follows from 3.3.20 since in both cases the bres considered there are regular rings see the proof of A.12. Proof.
Exercises
Let R be a ring, M an R -module, and R M the trivial extension of R by M . (The de nition of R M is given after Theorem 3.3.6.) Prove: (a) R M is a ring. (b) R can be identi ed with the subring R 0 = (a x) R M : x = 0 . (c) 0 M = (a x) R M : a = 0 is an ideal in R M with (0 M )2 = 0. As R -modules, M and 0 M are isomorphic. (d) If (R ) is local, then R M is local with maximal ideal M = (a x) R M: a . (e) The natural inclusion R R M composed with the natural epimorphism R M (R M )=(0 M ) is an isomorphism. (e) If R is Noetherian and M is a nite R -module, then R M is Noetherian and dim R = dim R M . 3.3.23. Let R be a Cohen{Macaulay local ring, C a maximal Cohen{Macaulay R -module, and x an R -sequence. If C=xC is the canonical module of R=(x), show that C is the canonical module of R . 3.3.24. Let (R ) be a Gorenstein local ring and I R an ideal of grade g such that S = R=I is a Cohen{Macaulay ring. Let x = x1 . . . xn be a system of generators of I . Show that !S = Hn g (x): 3.3.22.
f
2 g !
2
m
m
!
g
f
2
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m
;
g
f
2
119
3.3. The canonical module 3.3.25.
and let
Let (R ) be a Gorenstein local ring, I R a perfect ideal of grade g, m
0 ;;! Fg ;;! ;;! F0 ;;! 0 be a minimal free R -resolution of S = R=I . Prove: @ @g (a) The dual complex 0 ;;! F0 ;;! ;;! Fg ;;! 0 is acyclic, and Coker @g
= !S . (b) HomS (!S S )
= Ker(Fg S ! Fg 1 S )
= TorRg (S S ). R (c) S is Gorenstein if and only if Torg (S S )
= S . (If you nd this problem too di cult, consult 1.4.22 or 159], Section 7.5.) (d) Suppose g = 2 then (I ) = r(S ) + 1. In particular, if R is regular and S is Gorenstein, then S is a complete intersection 3.3.26. Let (R k) be a Gorenstein local ring of dimension d , and M a nite module of nite projective dimension. Show that @g
@1
1
;
m
TorRi (k M )
= ExtdR i (k M )
for all i.
;
Let (R ) be a Cohen{Macaulay local ring with canonical module !R . Suppose for all nite R -modules M there exist an integer n and an epimorphism !Rn ! M . Prove R is a Gorenstein ring. 3.3.28. Let (R ) be a Cohen{Macaulay local ring with canonical module !R . (a) Suppose M is a maximal Cohen{Macaulay R -module of nite injective dimension. Show M is isomorphic to a direct sum of nitely many copies of !R . (b) Let M be a nite R -module. Show inj dim M < 1 if and only if M has a nite !R -resolution, that is, there exists an exact sequence 3.3.27.
m
m
0 ;;! !Rrp ;;! ;;! !Rr 'p
'1
0
;;! M ;;! 0:
Hint: For all nite R -modules M there exists an exact sequence 0 ! Y ! X ! M ! 0 where X is a maximal Cohen{Macaulay R -module, and Y a module of nite injective dimension see 21]. Such an exact sequence is called a Cohen{Macaulay approximation. (c) The !R -resolution is minimal if Im 'i !Rri; for i = 1 . . . p. Show that a module M of nite injective dimension even has a minimal !R -resolution, and that ri = d i ( M ) for all i when the resolution is minimal. 3.3.29. Let (R ) be a Cohen{Macaulay local ring of dimension 1. A subset I of the total ring Q of fractions of R is called a fractionary ideal if there exist R -regular elements x, y such that y 2 xI R . The inverse of a fractionary ideal I is the set I 1 = fa 2 Q : aI R g. We denote by F the set of fractionary ideals of R . Show: (a) If I 2 F , then I 1 2 F and I (I 1 ) 1 . (b) If I R is a fractionary ideal, then `(R=I ) < 1, R I 1 and `(I 1 =R ) < 1. (c) The following conditions are equivalent: (i) R is a Gorenstein ring (ii) I = (I 1 ) 1 for all I 2 F, (iii) `(R=I ) = `(I 1 =R ) for all I 2 F, I R . 1
m
;
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m
;
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;
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120
3. The canonical module. Gorenstein rings
!
Let R S be a faithfully at homomorphism of Noetherian rings, and C a nite R -module. Show the following are equivalent: (a) C S is a canonical module of S (b) C is a canonical module of R , and for every prime ideal Spec S the bre S= S, = R , is Gorenstein.
3.3.30.
p
q
p
q
q
\
q
2
Let k be a eld, and R a k-algebra which is Cohen{Macaulay and admits a canonical module. Let K be a eld, and suppose that either R is a nitely generated k-algebra or K is a nitely generated extension eld of k. Show that !R k K is a canonical module of R k K . Hint: apply 2.1.11. 3.3.31.
3.4 Gorenstein ideals of grade 3. Poincare duality The Hilbert{Burch theorem 1.4.17 identi es perfect ideals of grade 2 as the ideals of maximal minors of certain matrices. For Gorenstein ideals of grade 3 there exists a similar `structure theorem' due to Buchsbaum and Eisenbud 65]. Let R be a Noetherian local ring. An ideal I R is a Gorenstein ideal (of grade g) if I is perfect and ExtgR (R=I R ) = R=I . Note that if R is Gorenstein and I is perfect, then I is Gorenstein if and only if R=I is Gorenstein. This follows from 3.3.7(b). To describe the structure theorem we recall a few facts from linear algebra: let R be a commutative ring, and F a nite free R -module. An R -module homomorphism ' : F ! F is said to be alternating if with respect to some (and therefore with respect to any) basis of F and the corresponding dual basis F , the matrix of ' is skew-symmetric and all its diagonal elements are 0. Suppose now that ' is alternating, choose a basis of F and the basis dual to this, and identify ' with the corresponding matrix (aij ). If rank F is odd, then det ' = 0, and if rank F is even, there exists an element pf(') 2 R , called the Pfaan of ', which is a polynomial function of the entries of ', such that det(') = pf(')2. For more details about Pfaans we refer the reader to 48], Ch. IX, x5, no. 2. We set pf(') = 0 if rank F is odd. Just like determinants, Pfaans can be developed along a row. Denote by 'ij the matrix obtained from ' by deleting the i-th and j -th rows and columns of ' then for all i, pf(') =
Xn
(;1)i+j ;1(i j )aij pf('ij )
j =1
((i j ) is the sign of j ; i). From now on we assume that the rank of F is odd, and consider the matrix derived from ' by repeating the i-th row
121
3.4. Gorenstein ideals of grade 3. Poincare duality
and column as indicated in the following picture 0 0 ai1 ain
B ;ai1 @ ...
=B B
'
;ain
1 CC CA :
Expansion with respect to the rst row of yields the equations 0 = ; pf() =
Xn j =1
(;1)j aij pf('j )
for i = 1 . . . n, where 'j is the matrix obtained from ' by deleting the j -th row and column. In other words, if we let : R ! F be the linear map de ned by (p1 . . . pn) (with respect to the given basis) where pj = (;1)j pf('j ), j = 1 . . . n, are the submaximal Pfaans of ', then we obtain the complex F. ('): 0
' ;! R ;! F ;! F ;! R ;! 0:
Theorem 3.4.1 (Buchsbaum{Eisenbud). Let (R ) be a Noetherian local m
ring. (a) Suppose n 3 is an integer, F a free R-module of rank n, and ' : F F an alternating map of rank n 1 whose image is contained in F . Then n is odd. Moreover, if Pf(') denotes the ideal generated by the submaximal Pfaans of ', then grade Pf(') 3. If grade Pf(') = 3, then F. (') is acyclic and Pf(') is a Gorenstein ideal. (b) Conversely, let I be a Gorenstein ideal of grade 3. Then there exist a free module F of odd rank and an alternating homomorphism ' : F F such that F. (') is a minimal free R-resolution of R=I. In particular, any Gorenstein ideal of grade 3 is minimally generated by an odd number of Pfaans.
;
!
m
!
Part (a) of the theorem is a consequence of the Buchsbaum{Eisenbud acyclicity criterion 1.4.13 and the following simple observation relating the ideal of (n ; 1)-minors of ' to the ideal of submaximal Pfaans. Lemma 3.4.2. Let (R ) be a Noetherian ring, F a free R-module of rank n, and ' : F ! F an alternating map of rank n ; 1. Then Rad Pf(') = Rad In;1('), and n is odd. Proof. For all i j = 1 . . . n, we denote by ij the matrix which is obtained from ' by deleting the i-th row and j -th column. Then In;1(') is generated by the elements det( ij ), and since pf('i)2 = det( ii) it follows right away that a power of Pf( ') is contained in In;1('). V Conversely, we consider the matrix of n;1 ' with entries det( ij ), m
122
3. The canonical module. Gorenstein rings
V
i j = 1 . . . n. It follows from Exercise 1.6.25 that rank n;1 ' = 1 since, by assumption, rank ' = n 1. Now 1.4.11 implies that all 2V minors of n;1 ' are zero. Therefore, since ' is skew-symmetric, we have det( ij )2 = det( ij ) det( ji) = det( ii) det( jj ) = pf('i)2 pf('j )2 for all i j = 1 . . . n. This implies that a power of In;1(') is contained in Pf('). Finally, since In;1(') = 0, we conclude that Pf(') = 0. This is only possible if n is odd.
;
;
6
6
For the proof of part (b) of 3.4.1 a little excursion to resolutions with algebra structures is needed. Let R be a commutative ring, and let P. :
;! P2 ;!d P1 ;!d P0 = R ;! 0
be an acyclic complex of projective modules. We may consider P as a graded module equipped with an endomorphism d : P ! P of degree ;1 satisfying d d = 0. The question is whether there can be de ned an associative multiplication on P satisfying the following rules: (a) PpPq Pp+q for all p q 0 (b) 1 2 P0 acts as the unit element, i.e. 1a = a1 = a for all a 2 P (c) ab = (;1)(deg a)(deg b)ba for all homogeneous elements a b 2 P (d) aa = 0 for all homogeneous elements a 2 P of odd degree (e) d (ab) = (da)b + (;1)deg aa(db) for all homogeneous elements a b 2 P . An example of a complex admitting such a multiplication is the Koszul complex. Unfortunately not all nite projective resolutions can be given an algebra structure with these properties. Avramov 25] found obstructions for this, and gave explicit examples of nite projective resolutions which fail to have such a structure. Nevertheless, if we do not insist on the associativity of the multiplication, we surprisingly have Theorem 3.4.3 (Buchsbaum{Eisenbud). Any projective resolution P with P0 = R admits a (possibly non-associative) multiplication satisfying the conditions (a){(e). Proof. We form the tensor product P P of complexes, and de ne the second symmetric power S2 (P ) of P to be .
.
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.
.
.
.
.
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.
.
P )=U where U is the graded submodule of P P which is generated by the elements a b ; (;1)(deg a)(deg b)b a with homogeneous a b 2 P , and the elements aa with homogeneous a 2 P of odd degree. Let d again denote the di erential of P P then d (U ) U . This implies that d induces S2 (P.) = (P.
.
.
.
.
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.
a di erential on S2 (P ), so that S2 (P ) inherits a complex structure. We .
.
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3.4. Gorenstein ideals of grade 3. Poincare duality
claim (and this is crucial for the proof) that the homogeneous components S2(P )k of this complex are all projective modules. Indeed, we have
M Pi Pj ) Tk
.
S2 (P.)k = (
8< 0 V Tk = : 2 Pk=2 S2 (Pk=2)
where
i+j =k i<j
if k is odd, if k is of the form 4n + 2, if k is of the form 4n. Thus S2 (P ) is a complex of projective R -modules which coincides with P in degrees 0 and 1. Therefore there exists a complex homomorphism : S2 (P ) ! P extending the identity in degrees 0 and 1, and we may assume that is chosen such that its restriction to R Pk is just the natural homomorphism to Pk . For all homogeneous elements a b 2 P we denote by ab the image of a b under the composition of the maps P P ;! S2 (P ) ;! P , and extend this multiplication by linearity to all other elements of P . It is clear that it has all the desired properties. Suppose now we are given a Noetherian local ring R and a Gorenstein ideal I R of grade g. Let F : 0 ;! Fg ;! ;! F1 ;! F0 ;! 0 be the minimal free resolution of R=I . The dual complex F is a minimal free resolution of ExtgR (R=I R ) = R=I , and hence must be isomorphic to F . Such an isomorphism is unique up to homotopy, and we are now choosing one which is derived from the multiplicative structure on F as given by 3.4.3. Observe that the multiplication de nes maps Fi Fg;i ! Fg = R which in turn induce R -module homomorphisms si : Fi ! Fg;i . For i = 0 . . . g we let s if i 0 1 mod 4, ti = ;i s if i 2 3 mod 4. i Proposition 3.4.4. t : F ! F is an isomorphism of complexes. In particular, si : Fi ! Fg;i is an isomorphism for i = 0 . . . g. Proof. We denote by d the di erential of F . Let a 2 Fi and b 2 Fg+1;i. Then ab = 0, and therefore 0 = d (ab) = d (a)b + (;1)i ad (b), or d (a)b = (;1)i+1ad (b). It follows that si;1(d (a))(b) = d (a)b = (;1)i+1ad (b) = (;1)i+1si (a)(d (b)) = (;1)i+1d (si(a))(b): .
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3. The canonical module. Gorenstein rings
Thus si;1 d = (;1)i+1d si which implies that t is a homomorphism of complexes. The induced homomorphism H0 (t ): R=I ! R=I must be an isomorphism since t0 = s0 is an isomorphism. Since t extends H0 (t ) it must be an isomorphism as well. We are now ready to prove 3.4.1(b): let .
.
.
F. : 0
.
' ;! F3 ;! F2 ;! F1 ;! R ;! 0
be the minimal free resolution of R=I equipped with a multiplication as in 3.4.3. Let e1 . . . en be a basis of F2 . Then, as we have just seen, s2(e1 ) . . . s2(en ) is a basis of F1 , and we may choose basis elements f1 . . . fn of F1 such that fi = s2 (ei ) for i = 1 . . . n. Then ei fj = ij g for all i j = 1 . . . n where g is a basis P element of F3 and ij denotes the Kronecker symbol. Let '(ei ) = nj=1 aij fj we claim that (aij ) is skew-symmetric, and all its diagonal elements are 0. To see this, notice that ej '(ei) = aij g. Therefore, aij (g) = (ej '(ei )) = '(ej )'(ei): The claim follows since is injective, and since '(ei)'(ej ) = ;'(ej )'(ei) and '(ei )'(ei) = 0 according rules. P to the multiplication Now let (g) = ni=1 ai ei since F = F , is isomorphic to the transpose of , and we conclude that I = (a1 . . . an). On the other hand, rank ' = n ; 1 and grade Pf(') = grade In;1('), by 3.4.2. Now grade In;1(') 3 since F becomes split exact after localizations at prime ideals of height 2. Thus part (a) of Theorem 3.4.1 implies that P F (') is acyclic. In particular it follows that Ker ' is generated by ni=1 pi ei where, up to signs, the pi are the submaximal Pfaans of '. Thus we have Pf(') = I , as desired. .
.
.
.
Poincare duality. Buchsbaum and Eisenbud 65] remark that the multiplication de ned on F. induces a multiplication on Tor. (k R=I ) giving
it the structure of an associative graded alternating algebra. They further point out that in view of 3.4.4, Tor (k R=I ) is a Poincare algebra if I is a Gorenstein L ideal. Recall that an associative graded alternating algebra A = gi=0 Ai is a Poincare algebra if for all i = 0 . . . g the A0homomorphisms Ai ! HomA (Ag;i Ag ), a 7! 'a with 'a(b) = ab, are isomorphisms. Notice that if R is regular, then there is a natural isomorphism between Tor (k R=I ) and the Koszul homology H (R=I ) = H (x R=I ) where x is a minimal set of generators of the maximal ideal of R see 1.6.9. It can be shown that this is an isomorphism of algebras. In particular, the Koszul homology H (R ) of a Gorenstein ring is a Poincare algebra. This is one direction of the theorem of Avramov and Golod 32] which asserts that a Gorenstein ring is characterized by its Koszul homology. Their theorem .
0
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.
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3.4. Gorenstein ideals of grade 3. Poincare duality
complements the result 2.3.11 of Tate and Assmus according to which the Koszul algebra of a complete intersection is an exterior algebra. We will present their proof which is independent of the above considerations. Theorem 3.4.5 (Avramov{Golod). Let (R k) be a Noetherian local ring, and let n = emb dim R ; depth R. The following conditions are equivalent: (a) R is a Gorenstein ring (b) H (R ) is a Poincare algebra (c) the k-linear map Hn;1 (R ) ! Homk (H1 (R ) Hn(R )) induced by the multiplication on H (R ) is a monomorphism. We begin with a few preliminary remarks. Suppose t = depth R > 0. By 1.2.2 (choose M = R and N = ), there exists an R -regular element y1 2 n 2 . Hence by induction on t we may construct an R -sequence y = y1 . . . yt such that y is part of a minimal system of generators of . By 1.6.13, one has H (R ) = H (R=y R ) as graded k-vector spaces. Inspecting this isomorphism we see that it is actually a k-algebra isomorphism. On the other hand, R is Gorenstein if and only if R=y R is too. Thus we may assume that depth R = 0. Let x = x1 . . . xn be a minimal system of generators of , and K = K (x) the Koszul complex of this sequence then, by de nition, H (R ) = H (K ). Note that K is a Poincare algebra: let e1 . . . en be an R basis of K1 then, in the terminology of Section 1.6, the eI , I f1 . . . ng, jI j = i, form an R-basis of Ki , and we have eI ^ eJ = (I J )e1 ^^ en for J f1 . . . ng, jJ j = n ; i. Here (I J ) = 1 if I \ J = , and 0 otherwise. This clearly proves that the maps !i : Ki ! HomR (Kn;i Kn ), !i (a) = 'a with 'a (b) = a ^ b, are isomorphisms as asserted. We denote by d the di erential of K . Then d and its dual anticommute with ! . In other words, we have !i;1 di = (;1)i;1 Hom(dn;i+1 Kn ) !i for all i = 0 . . . n. This equation is stated in 1.6.10, the only di erence being that there Kn is identi ed with R . It follows that the isomorphisms !i induce isomorphisms ! e i : Hi(R) ! H n;i(R) where we identify H n;i(R) n ; i with H ((K ) ), and where (K ) = HomR (K Kn ). Consider the diagram m
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m
m
m
m
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.
m
. .
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.
;;;;! Homk (Hn;i (R ) Hn(R )) ?? ??
Hi (R )
i
~y yi i H n;i (R ) ;;;;! HomR (Hn;i(R ) Kn ) Here the upper map i is induced by the multiplication on H (R ). We have just seen that !e i is an isomorphism. The lower map i is the natural homomorphism which assigns to a homology class , an (n ; i)-cycle in !i
.
126
3. The canonical module. Gorenstein rings
(K ), the (well de ned) homomorphism i ( ) 2 HomR (Hn;i(R ) Kn) with i( )( a) = (a), a 2 Kn;i a cycle. Next note that Hn (R ) = Soc Kn Kn . We de ne i to be Hom(Hn;i(R ) ) where : Soc Kn ! Kn is the natural inclusion. It is clear that i is an isomorphism. In fact, since Hn;i(R ) is annihilated by , any homomorphism Hn;i (R ) ! Kn necessarily maps Hn;i(R ) into Soc Kn . We leave it to the reader to check the commutativity of the diagram. In conclusion we have that i is a mono-, epi-, or isomorphism if and only if i is too. We will determine the kernel and cokernel of i. Lemma 3.4.6. Let B denote the boundaries of K . Then for any i we have .
m
.
.
a long exact sequence
n;i 0 ;! Ext1R (Ki;1 =Bi;1 Kn ) ;! H i (R ) ;;! HomR (Hi (R ) Kn) 1 ;! ExtR (Bi;1 Kn) ;!
The short exact sequence 0 ! Hi (R ) ! Ki =Bi ! Bi;1 ! 0 gives rise to the long exact sequence Proof.
i 0 ;! HomR (Bi;1 Kn ) ;! HomR (Ki =Bi Kn ) ;! HomR (Hi (R ) Kn) 1 ;! ExtR (Bi;1 Kn) ;! It is immediate to see that Im n;i = Im i, so that the sequence
H i (R )
;;! HomR (Hi (R ) Kn) ;! Ext1R (Bi;1 Kn ) ;! n;i
is exact. The module U of i-cycles of (K ) whose homology classes belong to Ker n;i is the module of homomorphisms 2 (Ki ) for which jZi = 0 (Zi cycles in Ki ). Therefore U is isomorphic to HomR (Bi;1 Kn ). Under this identi cation Ker n;i equals U=V where V is the module of homomorphisms : Bi;1 ! Kn which can be extended to Ki;1 . This means that Ker n;i = Ext1R (Ki;1=Bi;1 Kn ). In order to complete the proof of 3.4.5 we need the following Lemma 3.4.7. Let (R k) be a Noetherian local ring of depth 0. If Ext1R (k R ) = 0, then R is Gorenstein. Proof. The hypothesis implies that the functor HomR ( R ) is exact on the category of R -modules of nite length. This yields `(HomR (M R )) = `(M ) `(HomR (k R )) for any R -module of nite length M . Now assume dim R > 0. Then `(R= n ), and so `(HomR (R= n R )), tends to in nity with n. On the other hand, HomR (R= n R ) = 0 : n. 2 Since 0 : 0 : is an ascending chain of ideals, and since R .
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is Noetherian, this chain stabilizes. Consequently, `(HomR (R= n R )) is bounded, a contradiction. Thus R is a zero dimensional ring for which HomR ( R ) is an exact functor. Hence R is an injective R -module, and so R is Gorenstein by de nition. End of the proof of 3.4.5: We have already accomplished the reduction to the case depth R = 0. (a) ) (b): Since R is Gorenstein and Kn = R , all Ext groups in the exact sequence 3.4.6 vanish, and so i is an isomorphism for all i. (b) ) (c) is trivial. (c) ) (a): By assumption n;1 is injective, and this implies that n;1 is injective. Thus it follows from 3.4.6 that Ext1R (K0=B0 R ) = 0. Now 3.4.7 completes the proof since K0 =B0 = k. Corollary 3.4.8. Let R be a Gorenstein local ring which is not a complete intersection. Then H1 (R )n;1 = 0 for n = emb dim R ; dim R. Proof. Suppose the vector subspace H1 (R )n;1 of Hn;1(R ) is not zero. Then H1 (R )n 6= 0 since 1 : H1 (R ) ! Homk (Hn;1 (R ) Hn(R )) is an isomorphism. This contradicts 2.3.15. m
Exercises 3.4.9. Let k be a eld and I
k
X1 X2 X3 ]] the ideal generated by the polynomials X12 X22 , X12 X32 , X1 X2 , X1 X3 , X2 X3 . By 3.2.11 it is a Gorenstein ideal of grade 3. Compute its free resolution (as a k
X1 X2 X3 ]]-module). 3.4.10. Let (R k) be a Cohen{Macaulay local ring with canonical module !R , and x a minimal set of generators of . We denote by H. (M ) the Koszul homology of an R -module M with respect to x. Recall that H. (M ) is an H. (R )module. Let n = embdim R dim R . Show that for all i, 0 i n, the k-linear map Hi (R ) Homk (Hn i (!R ) Hn (!R )) which is induced by the scalar multiplication of H. (R ) on H. (!R ) is an isomorphism.
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3.5 Local cohomology. The local duality theorem The canonical module was introduced by Grothendieck in connection with the local duality theorem which relates local cohomology with certain Ext functors. We will describe this approach to the canonical module in this section. First local cohomology functors will be introduced, and it will be shown that the depth and the dimension of a module can be expressed in terms of their vanishing and non-vanishing. We end with the local duality theorem. Let (R k) be a Noetherian local ring and M an R -module. Denote by ; (M ) the submodule of M consisting of all elements of M with m
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support in f g. That is, m
f 2 M:
= 0 for some k 0g: Let F = (Ik )k 0 be a family of ideals of R such that Ij Ik for all j > k . Then F de nes a topology on R see 270], Section 8. F gives the -adic topology on R if and only if for each Ik there is a j 2 N such that j Ik , and for each i there is an l 2 N such that Il i . It is clear that for any such family one has ; (M ) = fx 2 M : Ik x = 0 for some k 0g: Let x = x1 . . . xn be a sequence of elements in R generating an -primary ideal. We set xk = xk1 . . . xkn for all k 0: The family (xk ) gives the -adic topology on R , and so ; (M ) = x m
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f 2 M : (xk)y = 0 for some k 0g: Noting that HomR (R=I M ) = fx 2 M : Ix = 0g for any ideal I of R , we obtain natural isomorphisms ; (M ) = lim ;! HomR (R= k M) = lim ;! HomR (R=(xk) M): ; (M ) = y m
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Proposition 3.5.1. ; ( ) is a left exact additive functor. Proof. The additivity of ; ( ) is trivial. We show that ; ( ) is left exact. If
0 ;! M1 ;! M2 ;! M3 m
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is exact, then we have a sequence 0 ;! ; (M1 ) ;! ; (M2) ;! ; (M3), where 0 = ; ( ) = j; (M ) and 0 = ; ( ) = j; (M ) . It is obvious that 0 is injective. Let x 2 Ker 0 then (x) = 0, and so there exists y 2 M1 such that x = (y). Since x 2 ; (M2), there exists an integer k 0 such that k x = 0. It follows that k (y) = ( k y) = 0. But is injective, and so y 2 ; (M1) and 0 (y) = x. De nition 3.5.2. The local cohomology functors, denoted by H i ( ), are the right derived functors of ; ( ). In other words, if I is an injective resolution of the R -module M , then H i (M ) = H i (; (I )) for all i 0. Remarks 3.5.3. (a) Let M be an R -module then H 0 (M ) = ; (M ) and H i (M ) = 0 for i < 0. (b) If I is an injective R -module, then H i (I ) = 0 for all i > 0. (c) For any R -module M and all i 0 one has 0
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H i (M ) = lim ExtiR (R= k M ) = lim ExtiR (R=(xk) M ) m
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where x is a sequence in R generating an -primary ideal. (d) A short exact sequence of R -modules 0 ;! M1 ;! M2 ;! M3 ;! 0 gives rise to a long exact sequence 0 ;! ; (M1 ) ;! ; (M2) ;! ; (M3) ;! H 1 (M1) ;! ;! H i;1(M3) ;! H i (M1) ;! H i (M2) ;! Only (c) needs some explanation: ; lim is an exact functor see 318], ! Theorem 2.18. Therefore if I is an injective resolution of M , then k H i (M ) H i (HomR (R= k I )) = H i (lim = lim R (R= I )) ;! Hom ; ! = lim ;! ExtiR (R= k M): Note that E(k) if = , ; (E (R= )) = 0 otherwise m
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see 3.2.7 and part (1) of the proof of 3.2.13. Using the structure of the minimal injective resolution E (M ) of M given in 3.2.9, we conclude that ; (E (M )) is a complex of the form 0 ;! E (k) ( M ) ;! E (k) ( M ) ;! ;! E (k)i ( M ) ;! This entails Proposition 3.5.4. Let (R k) be a Noetherian local ring and M a nite .
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R-module. (a) The modules H i (M ) are Artinian. (b) One has H i (M ) = 0 if and only if i < depth M. (c) If R is Gorenstein, then n H i (R ) = E (k) for i = dim R, 0 otherwise. ^ (d) Let N denote the -adic completion of an R-module N. Then ^) H i (M ) = H i (M ) R R^ = H i^ (M for all i 0: m
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(a), (b) and (c) follow from the structure of ; (E (M )) and the fact that depth M = inf fi : i( M ) 6= 0g. (d) As H i (M ) is Artinian, it is the direct limit of submodules Uj of nite length. For each Uj one has Uj R R^ = Uj , and so H i (M ) = lim ;!(Uj R R^ ) = (lim ;! Uj ) R R^ = H i (M) R R:^ Using the R -atness of R^ , we get ^ ^ j M^ ) H i (M ) R R^ = lim ;!i ExtiR (R= j M) R R^ = lim ;! ExtiR^ (R= = H ^ (M^ ): Proof.
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Local cohomology and the Koszul complex. Our next goal is to construct a more explicit complex whose cohomology gives us H . (M ). Let x = x1 . . . xn be a system of parameters of R . For all l 0 we get a m
commutative diagram
' ;;;;! K1 (xl ) ?? ?? (l ) 1
K1 (xl+1)
y
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K0 (xl )
with '(1l) (ei ) = xi ei for i = 1 . . . n. (In both Koszul complexes we denote the natural basis of K1 = R n by e1 . . . en ). V (l) i (l ) Let 'i = '1 then '(l) : K (xl+1) ;! K (xl ) is a complex homomorphism see 1.6.8. We denote by 'l : K (xl ) ;! K (xl+1) the dual complex homomorphism. This can be done for each l , and so we obtain a direct system of complexes. Thus we may form the complex lim ;! K (xl ): .
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On the other hand, one de nes a complex C : 0 ;! C 0 ;! C 1 ;! ;! C n ;! 0 M Ct = Rxi xi xit .
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1 i1 d, (b) H t (M ) = 0 and H d (M ) = 0.
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We rst note the following rule which will be used several times in the proof of (a) and (b): Let ' : (R k) ! (R 0 0 k0 ) be a local ring homomorphism such that R 0 is an 0 -primary ideal. Then for any R 0-module M one has (3) H i (M ) = H i 0 (M ) for all i 0: Of course, on the left hand side of this formula M is considered as an R -module. In fact, if = (x) with x = x1 . . . xn and if x0 = '(x1) . . . '(xn), then C R M = C 0 R0 M , where C and C 0 are the complexes of 3.5.6 de ned with respect to x and x0 . The isomorphism (3) follows from 3.5.6. (a) We only need to prove that H i (M ) = 0 for i > d . The other part of statement (a) has already been shown in 3.5.4. Let R ! R 0 = R= Ann M be the canonical epimorphism. Then M is an R 0-module with dim M = dim R 0. Using (3) we may therefore assume that dim R = dim M = d . Let x = x1 . . . xd be a system of parameters of R , and let C be the complex 3.5.6 de ned with respect to x. Then C i = 0 for i > d , and so H i (M ) = H i (M R C ) = 0 for i > d see 3.5.6. (b) We proceed by induction on t in order to show that H t (M ) 6= 0. If t = 0, then 0 6= Soc M H 0 (M ). Now suppose t > 0 then there exists an M -regular element x 2 . The exact sequence x 0 ;! M ;! M ;! M=xM ;! 0 yields the exact sequence 0 = H t;1 (M ) ;! H t;1 (M=xM ) ;! H t (M ). By our induction hypothesis we have H t;1 (M=xM ) 6= 0 this implies H t (M ) 6= 0. Proof.
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Finally we show that H d (M ) 6= 0. Using 3.5.4 and the fact that dim M^ = dim M for the -adic completion M^ of M , we may assume that R is complete. Let 2 Supp M with dim M = dim R= . Then dim M= M = dim M = d , and we get an exact sequence of R -modules 0 ;! U ;! M ;! M= M ;! 0 m
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inducing the exact sequence H d (M ) ;! H d (M= M ) ;! H d +1(U ). According to (a) we have H d +1(U ) = 0, and so, if H d (M= M ) 6= 0, then H d (M ) 6= 0. As M= M is an R= -module we may as well assume, by (3), that R is a domain and dim R = dim M . Any complete Noetherian domain has a Noether normalization: there exists a regular local subring (S ) such that R is a nite S -module see A.22. In particular, the extension ideal R is -primary. Again using (3) we may replace R by S , and so may assume that R itself is regular. Let K be the fraction eld of R , and let : M ! K R M be the canonical homomorphism. We set U = Ker and N = Im . Then we obtain the exact sequence (4) 0 ;! U ;! M ;! N ;! 0 and, as a consequence of Exercise 1.4.18, an exact sequence (5) 0 ;! N ;! R s ;! W ;! 0 where s = rank M = rank N and consequently dim W < dim R = d . As dim W < d , (5) yields the exact sequence H d (N ) ;! H d (R s ) ;! H d (W ) = 0: p
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We have H d (R s ) = H d (R )s = E (k)s (see 3.5.4) and so H d (N ) 6= 0. Finally, from the exact sequence (4) it follows that H d (M ) 6= 0. The next theorem is known as the local duality theorem. Theorem 3.5.8 (Grothendieck). Let (R k) be a Cohen{Macaulay comm
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plete local ring of dimension d. Then for all nite R-modules M and all integers i there exist natural isomorphisms
H i (M ) = HomR (ExtdR;i(M !R ) E (k)) and ExtiR (M !R ) = HomR (H d ;i (M ) E (k)): m
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The rst isomorphisms result from the second by Matlis duality 3.2.13. For the proof of the second isomorphisms note that both sides vanish for i < 0 see 3.5.7. For i 0 we set T i( ) = HomR (H d ;i ( ) E (k)). It is clear that T 0( ) is a contravariant left exact functor which maps
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direct sums to direct products. Hence there exists an R -module C such that T 0( ) = HomR ( C ) see 318], Theorem 3.36. It follows that C = T 0(R ). As H d (R ) is an Artinian module, Matlis duality 3.2.13 implies that C is a nite R -module. In order to conclude the proof we will show that the functors T i( ) are the right derived functors of T 0( ), and that C = !R . Remark 3.5.3 implies immediately that the functors T i ( ) are strongly connected (see 318], p. 212). Thus the T i( ) are the right derived functors of T 0( ), once we have shown that T i(F ) = 0 for every free R -module F and all i 1. The functors T i ( ) map direct sums to direct products, and so it suces to show that T i(R ) = 0 for i 1, or equivalently that H i (R ) = 0 for i < d . This however follows from 3.5.7 since R is Cohen{Macaulay. Summing up we have (6) HomR (H i (M ) E (k)) = ExtdR;i(M C ) for all i and all R -modules M . Now 3.5.7 yields i = 0, and therefore Exti (k C ) k for i = d , H i (k) = k0 for = R for i > 0, 0 for i 6= d , by (6). Thus it follows from the remark after 3.3.1 that C = !R . Grothendieck's duality theorem has the following often applied variant: Corollary 3.5.9. Let (R k) be a Cohen{Macaulay local ring of dimension m
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d which is the homomorphic image of a Gorenstein local ring. Then R has a canonical module, and for all nite R-modules M and all integers i there exist natural isomorphisms
H i (M ) = HomR (ExtdR;i(M !R ) E (k)): m
For the proof we apply 3.5.4, 3.3.5, and Exercise 3.2.14. Then ^ !R^ ) E (k)) H i (M ) = H i^ (M^ ) = HomR^ (ExtdR^ ;i(M = HomR (ExtdR;i(M !R ) E(k)): Remark 3.5.10. Let (R k) be a complete local ring. The proof of 3.5.8 shows that the functor HomR (H d ( ) E (k)) is representable, even if R is not a Cohen{Macaulay ring. In other words, there exist a unique R -module KR (in the proof of 3.5.8 this module was denoted by C ) and a canonical isomorphism HomR (H d (M ) E (k)) = HomR (M KR )
Proof.
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for all R -modules M . Of course, KR = !R if R is Cohen{Macaulay. Even in the more general situation when the ring is not Cohen{Macaulay, the module KR is often called the canonical module of R . Its properties have been investigated by Aoyama 10]. Schenzel 329] has introduced the canonical module KM of an R -module M . The local duality theorem combined with 3.5.7 allows us to generalize 3.3.10(d). Corollary 3.5.11. Let (R k) be a Cohen{Macaulay local ring of dimenm
sion n with canonical module !R , and M a nite R-module of depth t and dimension d. Then (a) ExtiR (M !R ) = 0 for i < n d and i > n t, (b) ExtiR (M !R ) = 0 for i = n d and i = n t, (c) dim ExtiR (M !R ) n i for all i 0. ^ !R^ ) for all i 0, since (!R )b = Proof. We have ExtiR (M !R )b = ExtiR^ (M !R^ (see 3.3.5). Under completion depth and dimension of a module are preserved. We may therefore assume that R is complete, and so (a) and
; ; ;
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(b) follow from 3.5.7 and 3.5.8. To prove (c), we choose 2 Supp ExtiR (M !R ) such that dim ExtiR (M !R ) = dim R= = dim R ; dim R : (The last equality holds since R is Cohen{Macaulay see 2.1.3.) By the choice of we have 0 6= ExtiR (M !R ) = ExtiR (M !R ) and so (a) yields i dim R = n ; dim ExtiR (M !R ). p
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Exercises Let (R ) be a Noetherian local ring, and M a nite R -module. Prove: (a) If i 1 ( M ) = 0 and i( M ) 6= 0, then H i (M ) 6= 0. (b) Suppose inj dim M = 1 then i( M ) 6= 0 for all i > dim M . 3.5.13. Find a Noetherian local ring (R ) of dimension d and depth t with (a) H i (R ) 6= 0 for i = t . . . d , (b) H i (R ) = 0 for i 6= t and i 6= d . 3.5.14. Let (S k) be a complete Cohen{Macaulay local ring, (R k) a residue class ring of S , and M a nite R -module. Show that HomR (H i (M ) ER (k))
= HomS (H i (M ) ES (k)) and derive the following version of the local duality theorem: for all integers i there exist natural isomorphisms HomR (H i (M ) ER (k))
d = dim S: = ExtdS i (M !S ) Hint: See the rst step in the proof of 3.5.7, and use 3.1.6. 3.5.12.
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Let (R k) be a regular local ring of dimension d 2. Let E be a nite R -module which is locally free on the punctured spectrum of R . That is, E is free for all Spec R , = . Show (a) `(H i (E )) < for all i < d , (b) the R -dual E of E is again locally free on the punctured spectrum of R , and H i (E ) = 0 for i = 0 1, (c) H i+1 (E ) = HomR (H d i (E ) E (k)) for i = 1 . . . d 2. 3.5.15.
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3.6 The canonical module of a graded ring For a graded ring R we de ne the canonical module in the category of graded R -modules and establish the graded version of the local duality theorem. Under certain restrictive assumptions on R the degrees of the generators in a minimal set of generators of the canonical module are uniquely determined, and one de nes the a-invariant of R to be the smallest of these degrees, multiplied by ;1. We adopt the assumptions and notation of Section 1.5. Thus R will be a Noetherian graded ring, and M0(R ) the category of graded R -modules. M0(R) is an Abelian category which has direct sums and direct products see 1.5.19. Likewise limits and colimits exist in M0(R ). As we have already mentioned in Section 1.5, M0(R ) has enough projectives. Our next concern will be to show that M0(R ) has enough injectives as well. Injective modules.
A graded R -module M is called injective if it is an injective object in M0 (R ). One sees easily that this is the case if and only if the functor HomR ( M ): M (R ) ;! M (R ) 0 0 is exact. A injective module M need not be injective (in the category M(R ) see 3.6.5). Just as in the category of all R -modules, one calls an extension N M of graded R -modules essential if for any graded submodule 0 6= U M one has U \ N 6= 0. If, in addition, N 6= M , the extension is called a proper essential extension. Similarly as in the non-graded case (see 3.2.2) one shows Proposition 3.6.1. A graded module is injective if and only if it has no proper essential extension.
We now prove that any graded R -module has a injective hull. In analogy to the de nition in the non-graded case, E is called a injective hull of M if it is injective and a essential extension of M . Theorem 3.6.2. Any graded R-module M admits a injective hull, and any
two injective hulls of M are isomorphic.
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We embed M into a (not necessarily graded) injective R -module I . According to 3.1.8 this is possible. Similarly as in the proof of 3.2.4 we consider the set = N : M N I M N is essential . We de ne a partial order on by setting N1 N2 if N1 is a graded submodule of N2 . Zorn's lemma applied to this set yields a maximal essential extension M E with E I . Suppose E is not injective then E has a proper essential extension E E 0 by 3.6.1. As I is injective, there exists an R -module homomorphism ' : E 0 I (not necessarily homogeneous), extending the inclusion E I . We claim that ' is injective. In fact, assume that there is a non-zero element x Ker ', say x = xr + + xs , with xi homogeneous of degree i, r s, and xr = 0. We show by induction on s r that there exists a homogeneous element a R such that ax E and ax = 0. Since ' E is injective, this gives a contradiction. If s r = 0, x is homogeneous, and the assertion follows since the extension E E 0 is essential. Now suppose that s r > 0. We choose a homogeneous element a R such that axr E 0 . Let Proof.
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; 2 nf g x0 = x ; xr = xr+1 + + xs : If ax0 = 0, then ax = axr 2 E n f0g and we are done. Otherwise ax0 6= 0, and by our induction hypothesis we may choose a homogeneous element b 2 R such that bax0 2 E and bax0 = 6 0. Then bax = bax0 + baxr 2 E, and bax = 6 0. Next let Ee = Im '. As ' is injective we may give Ee a natural graded structure (Ee i = '(Ei) for all i 2 Z). Then E Ee is a proper essential extension with Ee I , contradicting the maximality of E . 2
The uniqueness of the injective hull is proved as in the non-graded case. We denote the injective hull of a graded R -module M by E (M ) or ER (M ) . The preceding theorem implies in particular that any graded R -module N has a injective resolution. That is, there exists a complex
;! I 0 ;! I 1 ;! I 2 ;! with injective modules I i such that H 0 (I ) = N and H i (I ) = 0 for i > 0. I. : 0
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Given such a injective resolution I of N we have .
Exti (M N ) = H i (HomR (M I . )) R
for all i 0 and all graded R -modules M . We omit the proofs of the following two results which have their analogues in 3.2.6, 3.2.7, 3.2.8, and 3.2.9, and which can be proved along the same lines as in the corresponding local case.
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Theorem 3.6.3. Let R be a Noetherian graded ring. Then (a) Ass E (M ) = Ass M for all M 2 M0(R ), (b) E 2 M0(R ) is a indecomposable injective module if and only if E = E (R= )(n)
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for some graded prime ideal R and some integer n Z, (c) every injective module can be decomposed into a direct sum of indecomposable injective modules, and this decomposition is unique up to homogeneous isomorphism. Proposition 3.6.4. Let R be a Noetherian graded ring, and M a graded R-module. Consider the minimal injective resolution p
0 ;! M ;! E 0 (M ) ;! E 1(M ) ;! of M (which is obtained recursively by setting E i (M ) = E (Im d i;1 )). Then, for every graded prime ideal of R and for every integer i 0, the Bass number i ( M ) equals the number of graded R-modules of the form E (R= )(n), n 2 Z, that appear in E i (M ) as direct summands. For any M 2 M0 (R ) we denote by inj dim M the injective dimension. Theorem 3.6.5. Let R be a Noetherian graded ring and M 2 M0 (R ). Then (a) inj dim M inj dim M + 1, (b) if M is injective, then inj dim M = 1 if and only if 2 Ass M for d0
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For the proof of 3.6.5 we will use Proposition 3.6.6. Let M 2 M0(R ) and a non-graded prime ideal in R. Then 0( M ) = 0, and i+1( M ) = i ( M ) for every integer i 0. The proof of this proposition is already given in 1.5.9, where we actually prove more than is stated in that theorem itself. Proof of 3.6.5. (a) We may assume that inj dim M = t < 1. Let 2 Spec R we want to prove that i( M) = 0 for i t + 2. This is certainly true when is a graded prime ideal see 3.6.4. Now suppose that is not a graded prime ideal. Then i+1( M ) = i ( M ) by 3.6.6 and i ( M ) = 0 for i t + 1, hence the assertion follows. (b) inj dim M = 1 happens if and only if 1( M ) 6= 0 for some nongraded prime ideal of R . But 1 ( M ) = 0 ( M ), and so 1 ( M ) 6= 0 if and only if 2 Ass M . Corollary 3.6.7. Let R be a Noetherian graded ring and a graded maximal p
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ideal of R. Then
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) = E (R= ): Proof. By 3.6.3 we have Ass E (R= ) = f g, and so 3.6.5 implies that E (R= ) is injective as an object in M(R ). Since 0 ( E (R= )) = 1 (see E (R=
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3.6. The canonical module of a graded ring
3.6.4), we conclude that E (R= ) is indecomposable in M(R ), and hence by 3.2.6 it must be isomorphic to E (R= ). m
m
The canonical module of a graded ring. Recall from 1.5 that a graded
ring is a local ring if it has a unique maximal ideal, that is, a graded ideal which is not properly contained in any graded ideal 6= R . De nition 3.6.8. Let (R ) be a Cohen{Macaulay local ring of dimension d . A nite graded R -module C is a canonical module of R if there exist homogeneous isomorphisms 0 for i 6= d , Exti (R= C ) = R= for i = d . R m
m
m
m
For a nite graded R -module M it may happen that there exists a homogeneous isomorphism M = M (i) with i 6= 0. To avoid this phenomenon, one has to require that R has no homogeneous units of positive degree. For a local ring (R ) this is the case if and only if is maximal (in the usual sense) see 1.5.16. Proposition 3.6.9. Let (R ) be a Cohen{Macaulay local ring, and C be m
m
a canonical module of R. Then (a) C is a canonical module of R, (b) C is uniquely determined up to homogeneous isomorphism, provided is maximal. Proof. (a) We need to show that C is a canonical module of R for all Spec R . First, let Spec R be a graded prime ideal then . The de nition of the canonical module implies that C is a canonical module of R , and so C is a canonical module of R see 3.3.5. Now let Spec R be a non-graded prime ideal. Then i+1( C ) = i ( C ), by m
m
p
2
p
2
p
p
p
m
m
2
m
p
p
3.6.6, and the assertion follows again. (b) Let C 0 be another canonical module. Remark 3.3.17 implies that HomR (C C 0) is a projective module of rank 1, and hence, as a graded module, is free (see 1.5.15). Therefore, HomR (C C 0) = R (i) for some i 2 Z. This implies HomR (C C 0(;i)) = R . Let ' 2 HomR (C C 0(;i)) be an element corresponding to 1 under this identi cation. Then, since HomR (C C 0(;i)) = HomR (C C 0) by Exercise 1.5.19(f), it follows from (a) and 3.3.4(c) that ' is locally an isomorphism. But then ' is a homogeneous isomorphism, and we have R= = ExtdR (R= C 0 (;i)) = ExtdR (R= C ) = ExtdR (R= C 0)(;i) = (R= )(;i): Therefore i = 0, and C = C 0. Example 3.6.10. Let R = kX1 . . . Xn] be a polynomial ring over a eld, and assign to the indeterminates the degree deg Xi = ai > 0 for i = 1 . . . n. p
p
m
m
m
m
m
p
140
3. The canonical module. Gorenstein rings
The maximal ideal of R is = (X1 . . . Xn), and the Koszul complex of X1 . . . P Xn yields a homogeneous free resolution of R= whose last term that Exti (R= R ) = 0 for i 6= n, is R (; ni=1 ai ). From this one concludes P n n and Ext (R= R ) = (R= )( i=1 ai ). In other words, the canonical P n module of R is R (; i=1 ai ). Proposition 3.6.11. Let (R ) be a Cohen{Macaulay local ring with m
m
m
m
m
canonical module !R . The following conditions are equivalent: m
(a) R is a Gorenstein ring (b) !R = R (a) for some integer a 2 Z. Proof. R is Gorenstein if and only if !R is locally free. By 1.5.15(d) this is the case if and only if !R = R (a) for some a 2 Z. The number a occurring in 3.6.11 is a numerical invariant of the Gorenstein local ring (R ), provided is maximal. In the case of a positively graded algebra over a eld it will be given a special name see 3.6.13. Let (R ) be a Cohen{Macaulay local ring with canonical module !R . The canonical module is a graded module, and by 1.5.15, every minimal system of homogeneous generators of !R has exactly ((!R ) ) elements. In analogy to the local case we de ne this number to be the type of R , and denote it by r(R ). In view of 3.6.11 it is clear that R is Gorenstein if and only if r(R ) = 1. For the sake of completeness we list a few change of rings properties of the canonical module. While part (a) of the next proposition follows easily from the results proved so far, it is best to use the change of rings spectral sequence Extp (k Extq (S !R )) =) Extn (k !R ) S R R p m
m
m
m
for (b) (see 318], 11.66) the reduction we used for the corresponding local result 3.3.7 is only possible if homogeneous systems of parameters are available. Proposition 3.6.12. Let (R ) be a Cohen{Macaulay local ring with canonical module !R .
m
(a) If is a graded prime ideal of R, then !R = (!R )( ) up to a shift. (b) Let ' : (R ) ! (S ) be a ring homomorphism of Cohen{Macaulay ( )
p
p
local rings satisfying m
p
n
(i) '(Ri) Si for all i 2 Z, (ii) '( ) , (iii) S is a nite graded R-module. Then !S = ExttR (S !R ), where t = dim R ; dim S. Example 3.6.10 and 3.6.12 imply that any Cohen{Macaulay positively graded algebra over a eld admits a canonical module. Following Goto and Watanabe 134] we de ne: m
n
141
3.6. The canonical module of a graded ring
De nition 3.6.13. Let k be a eld, and R a Cohen{Macaulay positively graded k-algebra. Then a(R ) = ; minfi : (!R )i 6= 0g is called the a-invariant of R . As a consequence of 3.6.12 we have Corollary 3.6.14. Let R be a Cohen{Macaulay local ring with canonical module !R , and let x = x1 . . . xn be an R-sequence of homogeneous elements with deg xi = ai for i = 1 . . . n. Then
!R=xR = (!R =x!R )(
Xn i=1
ai ):
In particular, if k be a eld, and RP a Cohen{Macaulay positively graded k-algebra, then a(R=xR ) = a(R ) + ni=1 ai . Proof. The Koszul complex ) is a graded free R -resolution of PnK.(ax). RFrom R=xR , and Kn (x R ) = R ( 3.6.12 we obtain i=1 i
;
Xn
!R=xR = ExtnR (R=xR !R ) = H n (x !R ) = (!R =x!R )(
i=1
ai):
Examples 3.6.15. (a) A graded polynomial ring R = kX1P . . . Xn] over a eld k with deg Xi = ai > 0 has the a-invariant a(R ) = ; ni=1 deg ai . (b) Let k be a eld, and R ! S a homomorphism of Cohen{Macaulay positively graded rings with maximal ideals and , respectively. Suppose the homomorphism satis es the conditions of 3.6.12(b), and suppose further that S has a nite free homogeneous R -resolution 0 ;! Ft ;! Ft;1 ;! ;! F0 ;! S ;! 0 L where t = dim R ; dim S . Write Ft = i2Z R (;ai ) then a(S ) = a(R ) + maxfai : i 2 Zg. This is proved exactly as in the special case 3.6.14. m
n
Local Duality. Our nal objective is to derive the graded version of the
local duality theorem. We begin with Matlis duality. Let (R ) be a Noetherian local ring then R0 is local with maximal ideal 0. We consider R0 as a graded ring by de ning (R0)i = 0 for i 6= 0. Similarly any R0-module may be considered a graded R0 -module concentrated in degree 0. Moreover, if M is a graded R -module, it may be viewed as a graded R0-module as well. Thus we can de ne M _ = HomR (M ER (R0= 0)): A priori, M _ is a graded R0 -module whose grading is given by (M _ )i = HomR (M;i ER (R0= 0)) m
m
0
0
0
m
0
m
142
3. The canonical module. Gorenstein rings
for all i 2 Z. But it is obvious that M _ has a natural structure as a graded R -module. The Noetherian local ring (R ) is said to be complete if (R0 0 ) is complete. If (R ) is complete and M is a nite graded R -module, then all homogeneous components Mi of M are complete R0-modules (since they are nite R0-modules). Proposition 3.6.16. Let (R ) be a Noetherian complete local ring. Then (a) the additive contravariant functor ( )_ : M0 (R ) ;! M0(R ) is exact (b) M _ = HomR (M R _ ) for all graded R-modules M (c) one has R _ = ER (R= ). Proof. (a) is obvious. (b) We de ne ' : HomR (M HomR (R E )) ! HomR (M E ) by setting '( )(x) = (x)(1) for all 2 Hom(M HomR (R E )) and all x 2 M . It is readily seen that ' is an isomorphism. (c) It follows from (a) and (b) that HomR ( R _ ) is an exact functor, and so R _ is injective. R _ is indecomposable since R __ = R . Note further that (R= )_ = R= . This is clear in the case where R= =k ; 1 is a eld, and it is easy to see in the case R= k t t ], since then = all homogeneous components of kt t;1 ] are isomorphic to k. Therefore the canonical epimorphism R ! R= yields a monomorphism R= = (R= )_ ;! R _ , and the assertion follows from 3.6.3. M 2 M0(R ) is called Artinian if every descending chain of graded submodules terminates. The homogeneous socle of a graded R -module M is de ned to be Soc M = HomR (R= M ). It is an R= -module and can be viewed as a graded submodule ofLM . As an R= -module it is free (see Exercise 1.5.20), and so Soc M = i2I (R= )(ai). If M is Artinian, then Soc M can have only nitely many summands (R= )(ai). Hence we may write n M Soc M = (R= )(ai ): m
m
m
m
m
0
0
0
m
m
m
m
m
m
m
m
m
m
m
m
m
i=1
As in the proof of 3.2.13 we conclude that M is Artinian if and only if there exist an integer n 0 and integers a1 . . . an such that (7)
M
M R_(ai): n
i=1
Let A0(R ) denote the full subcategory of M0 (R ) consisting of all Artinian R -modules and F0(R ) the full subcategory of all nite graded R -modules. Theorem 3.6.17 (Matlis duality for graded modules). Let (R ) be a Noetherian complete local ring, and let M 2 F0 (R ) and N 2 A0(R ). m
Then
143
3.6. The canonical module of a graded ring
(a) M _ 2 A0(R ) and N _ 2 F0(R ), (b) M __ = M and N __ = N, (c) the functor ( )_ : F0(R ) ;! A0(R ) establishes an anti-equivalence of categories.
Using (7) one proves the theorem in the same way as 3.2.13. For example, in order to show (b) we set E = ER (R0= 0 ). Then we have Proof.
0
(M __ )i = HomR (HomR (Mi E ) E ) = Mi 0
m
0
by Matlis duality see 3.2.13. Now let (R ) be a Noetherian local ring. For M 2 M0(R ) we de ne m
H i (M ) = lim Exti (R= R
;!
m
k M
m
)
it is called the i-th local cohomology functor. H 0 ( ) is left exact and the functors H i ( ), i 0, are the right derived functors of H 0 ( ). Remark 3.6.18. Assume in addition that the maximal ideal of R is maximal. Since for all i and j , and all M 2 M0 (R ), we have Exti (R= j M ) = ExtiR (R= j M ) = ExtiR (R = j R M ), we see that R in this case H i (M ) = HRi (M ). Theorem 3.6.19 (The local duality theorem for graded modules). Let (R ) be a Cohen{Macaulay complete local ring of dimension d. Then (a) R has a canonical module !R , and !R = (H d (R ))_ , (b) for all nite graded R-modules M and all integers i there exist natural m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
homogeneous isomorphisms
(H i (M ))_ = ExtdR;i (M !R ): m
The proof follows as in the non-graded case see 3.5.8. Let R be a positively graded k-algebra. Then it follows from 3.6.19(a) that a(R ) = maxfi : H d (R )i 6= 0g: If in addition dim R = 0, then a(R ) = maxfi : Ri 6= 0g. m
Exercises Let R be a Noetherian graded ring. (a) For 2 Spec R show that R is Gorenstein if and only if R is Gorenstein. (b) Show the following conditions are equivalent: (i) R is a Gorenstein ring (ii) R is a Gorenstein ring for all graded prime ideals 2 Spec R (iii) R( ) is a Gorenstein ring for all graded prime ideals 2 Spec R . 3.6.20.
p
p
p
p
p
p
p
144
3. The canonical module. Gorenstein rings
(c) Let (R ) be local ring. Deduce that R is Gorenstein if and only if R is Gorenstein. 3.6.21. The purpose of this exercise is to re-prove a few results of Goto and Watanabe. L Let R be a graded ring, d a positive integer. The ring R (d ) = i Z Rid is called the d-th Veronese subring of R . It is a graded subring of R with grading (d ) ( R (d ) ) i = LRid for all i 2 Z. For j = 0 . . . d ; 1 we consider the graded R -modules Mj = i Z Rid +j with grading (Mj )i = Rid +j for all i 2 Z. We assume that R is Noetherian, and for (d) and (e) that it is a positively graded algebra over a eld. Show: L (a) R = dj=01 Mj (as R (d ) -module). In particular, R (d ) is a direct summand of R R (d ) is Noetherian, and the Mj are nite R (d ) -modules. (Hint: Compare the proof of 1.5.5.) (b) R is Cohen{Macaulay if and only if all Mj are maximal Cohen{Macaulay R (d ) -modules. L (c) If R is Cohen{Macaulay, then !R d
= i Z(!R )id . (d) If R is Gorenstein and a(R ) = bd + j , 0 j d ; 1, then !R d
= Mj (b). (e) If R is Gorenstein and a(R ) 0 mod d , then R (d ) is Gorenstein. Is a(R ) 0 mod d if R and R (d ) are Gorenstein? 3.6.22. Let k be a eld. We consider R = k X1 . . . Xn ] as a graded k-algebra with deg Xi = ai > 0 for i = 1 . . . n. Determine all Veronese subrings of R which are Gorenstein. 3.6.23. Let R be a homogeneous k-algebra, k a eld. Express a(R (d ) ) in terms of a(R ) and d .
m
m
2
2
;
( )
2
( )
Notes Grothendieck introduced the canonical module (often called dualizing module) and proved the local duality theorem. A comprehensive presentation of this theory including local cohomology is given in 143]. Equally fundamental is the famous paper of Bass 37]. The interested reader can nd some more historical background there. We were guided by the books of Kaplansky 231] and Matsumura 270] in Sections 3.1 and 3.2. In Sections 3.3 and 3.5 we follow partly the lecture notes of Herzog and Kunz 159]. Part of Section 3.5 has been inuenced by the notes of P. Roberts 311]. In particular the description % complex has been taken from this source. In Section of the modi ed Cech 3.6 we follow to a large extent the papers 134] by Goto and Watanabe and 109] by Fossum and Foxby. The main result 3.1.17 of Section 3.1 is due to Bass 37]. The characterization 3.2.10 of Gorenstein rings in terms of the type was rst proved by Bass 36]. Bass 37] gives a list of other equivalent conditions for the Gorenstein property. In 115] Foxby proves the following conjecture of Vasconcelos, which is a remarkable characterization of Gorenstein rings. Suppose (R k) is a Noetherian local ring of dimension d containing a m
145
Notes
eld then R is a Gorenstein ring if d ( R ) = 1. (We will give a proof of this result in 9.6.3.) The main theorem of Matlis duality and the structure theorem for injective modules are proved by Matlis in 269], and can also be found in the more general framework of Abelian categories in Gabriel 122]. Theorem 3.3.10 which characterizes the canonical module is taken from 159]. However in the proof given here we do not use local cohomology. Theorem 3.3.6 on the existence of the canonical module is independently due to Foxby 113] and Reiten 307], and 3.3.19 is a theorem of Murthy 281]. It says that every factorial Cohen{Macaulay ring with a canonical module is Gorenstein. In 372] Ulrich proves a certain converse of Murthy's theorem: any Gorenstein ring which is a factor ring of a regular local ring and which is locally a complete intersection in codimension one can be realized as a specialization of a Cohen{Macaulay factorial domain. For a while it had been open whether or not there exist non{Cohen{ Macaulay factorial local rings. Such examples were found by Bertin 41] (see 6.4.7) and also by Fossum and Grith 111]) in characteristic p and by Freitag and Kiehl 118] in characteristic 0. There are two remarkable extensions of the theory of the canonical module in its basic form as presented here: Sharp introduced Gorenstein modules in 337] as those nite modules G whose Cousin complex provides a (minimal) injective resolution for G. A Gorenstein module shares many properties with the canonical module. It is a Cohen{Macaulay module of nite injective dimension whose type and rank, however, may be bigger than one. We refer the reader to the papers on Cousin complexes and Gorenstein modules 336], 337], 338], and 341] by Sharp, and the article 113] by Foxby. It is shown in 339] that for a Noetherian local ring admitting a canonical module !R , any Gorenstein module is a direct sum of copies of !R . However there exist Cohen{Macaulay local rings not admitting a canonical module. A rst example of a one dimensional ring with this property was given by Ferrand and Raynaud 106], and an example of a factorial Cohen{Macaulay ring without a canonical module is due to Ogoma 294]. In 392] Weston gives an example of a ring with a Gorenstein module of rank 2, admitting no canonical module. The second extension of the basic concept gives a duality theory even for non{Cohen{Macaulay rings. In this theory the canonical module has to be replaced by the so-called dualizing complex, and duality is obtained in the derived category. We refer the reader to the book of Hartshorne 151]. A more elementary account of the theory can be found in Sharp 340]. As a consequence of the structure theorem 3.4.1 for Gorenstein ideals of grade three, these ideals have an odd number of generators. This had been observed before by J. Watanabe 386]. He uses linkage arguments m
146
3. The canonical module. Gorenstein rings
in his proof. Linkage had already been considered in 1945 by Apery 11] and by Gaeta 123] in 1952. It has become popular as a result of the paper of Peskine and Szpiro 298]. Linkage provides a technique to construct large and interesting classes of perfect ideals or of Gorenstein ideals whose structure is well understood. Of particular interest are the ideals in the linkage class of a complete intersection, called licci ideals. The simplest examples are the so-called Northcott ideals 288] and the Gorenstein ideals de ned in 156]. More important is the fact that perfect ideals of grade two 298] and Gorenstein ideals of grade three 386] are in the linkage class of a complete intersection. They are in a sense the archetypes of licci ideals as shown by Huneke and Ulrich 220]. For further study of linkage theory we refer the reader to the papers of Huneke 210], 213], Huneke and Ulrich 219], 221], 222], Kustin and Miller 251], 252], 254], and Ulrich 374], 373]. The height 3 monomial Gorenstein ideals have been completely classi ed by Bruns and Herzog 58]. There have been attempts to obtain structure theorems for nonGorenstein ideals of grade 3 or even for ideals of grade higher than 3. The next case of interest is Gorenstein ideals of grade 4. As a rst approach to the problem one may try to classify the Tor-algebras of these ideals I , i.e. TorR (k R=I ) when I is an ideal in the local ring R with residue class eld k. For ideals I such that proj dim R=I = 3 this has been done by Weyman 393] in characteristic 0 and by Avramov, Kustin, and Miller 33] in all characteristics. The next case of interest is Gorenstein ideals of grade 4. At the moment a general structure theorem for these ideals seems to be out of range. Kustin and Miller 253] succeeded in classifying their Tor-algebras. A remarkable result, valid for ideals of arbitrary grade, is due to Kunz 246]: if I is a Gorenstein ideal, then (I ) 6= grade I + 1. Duality theory is a classical and fundamental topic in algebraic geometry, and has also several algebraic aspects we have not even touched upon. We must content ourselves with a list of keywords and references: Riemann{Roch theorem, Serre duality, modules of regular di erentials, residue symbols, trace maps see Hartshorne 151], 152], Kunz 247], Kunz and Waldi 250], Lipman 259], Scheja and Storch 324], 325], and Serre 333].
4 Hilbert functions and multiplicities
The Hilbert function H (M n) measures the dimension of the n-th homogeneous piece of a graded module M . In the rst section of this chapter we study the Hilbert function of modules over homogeneous rings, prove that it is a polynomial for large values of n, and introduce the Hilbert series and multiplicity of a graded module. The next section is devoted to the proof of Macaulay's theorem which describes the possible Hilbert functions. The third section complements these results by Gotzmann's regularity and persistence theorem. The Hilbert function behaves quite regular, even for graded, nonhomogeneous rings. Such rings will be considered in the fourth section, where we will also investigate the Hilbert function of the canonical module. The passage to the associated graded ring with respect to a ltration allows us to extend some concepts for graded rings like `Hilbert function' or `multiplicity' to non-graded rings, and leads to the Hilbert{Samuel function and the multiplicity of a nite module with respect to an ideal of de nition. We shall study basic properties of ltrations and their associated Rees rings and modules, and sketch the theory of reduction ideals. Finally we prove Serre's theorem which interprets multiplicity as the Euler characteristic of a certain Koszul homology. 4.1 Hilbert functions over homogeneous rings We begin by studying numerical properties of nite graded modules over a graded ring R . Our standard assumption in this section will be that R0 is an Artinian local ring, and that R is nitely generated over R0 . Notice that for each nite graded R -module M , the homogeneous components Mn of M are nite R0-modules, and hence have nite length. De nition 4.1.1. Let M be a graded R -module whose graded components Mn have nite length for all n. The numerical function H (M ): Z ! Z with n) = `(Mn) for all n 2 Z is the Hilbert function, and HM (t) = P HH((M M n)tn is the Hilbert series of M . n2Z For the rest of this section we will assume that R is generated over R0 by elements of degree 1, that is, R = R0 R1]. Recall that such a ring is
said to be homogeneous.
147
148
4. Hilbert functions and multiplicities
! 2
We say that a numerical function F : Z Z is of polynomial type (of degree d ) if there exists a polynomial P (X ) QX ] (of degree d ) such that F (n) = P (n) for all n 0. By convention the zero polynomial has
degree ;1. We de ne the dierence operator on the set of numerical functions by setting (F )(n) = F (n + 1) ; F (n) for all n 2 Z. Notice that maps polynomial functions to polynomial functions, lowering the degree of non-zero polynomials by 1. The d times iterated -operator will be denoted by d . We further set 0F = F . Lemma 4.1.2. Let F : Z ! Z be a numerical function, and d 0 an integer. The following conditions are equivalent: (a) d F (n) = c, c = 0, for all n 0 (b) F is of polynomial type of degree d.
6
(b) ) (a) is easy. We prove the other implication by induction on d . The assertion is trivial for d = 0. Now assume that d > 0, and d F (n) = d ;1(F (n + 1) ; F (n)) = c, c 6= 0, for all n 0. By the induction hypothesis it then follows that there exist an integer n0 and a polynomial P (X ) 2 QX ] of degree d ; 1 such that P F (n + 1) ; F (n) = P (n) for all n n0 . Then F (n + 1) = F (n0) + nk=n P (k), and this last sum is a polynomial function in n of degree d . After these preparations we can state the main result of this section as follows: Theorem 4.1.3 (Hilbert). Let M be a nite graded R-module of dimension d. Then H (M n) is of polynomial type of degree d ; 1. Proof. We prove the theorem by induction on the dimension d of M . First note that there is a chain 0 = N0 N1 Nn = M of graded submodules of M such that for each i we have Ni+1=Ni = (R= i )(ai) where i is a graded prime ideal. Indeed, we may assume that M 6= 0. Choose 1 2 Ass(M ). The prime ideal 1 is graded see 1.5.6. There exists a graded submodule N1 M with N1 = (R= 1 )(a1). If N1 6= M we choose 2 2 Ass(M=N1 ). Then there exists a graded submodule N2 M with N2 =N1 = (R= 2 )(a2). If N2 6= M , we may proceed in the same way. But M is Noetherian, and so this process terminates eventually. Now, since the Hilbert P function is additive on short exact sequences, it follows that H (M n) = i H ((R= i)(ai ) n). Notice that d is the supremum of the numbers dim R= i . Hence the theorem will follow once we have shown it for M = R= , a graded prime ideal. (Here, of course, one has to observe that the polynomials describing Hilbert functions are zero or have positive leading coecients since their values are non-negative for n 0. As a consequence, the degree of the sum of such polynomials is the maximum of their degrees.) Proof.
0
p
p
p
p
p
p
p
p
p
p
p
149
4.1. Hilbert functions over homogeneous rings
R= = 0, then is the unique graded maximal ideal 0 L If Rdim n of R , where 0 is the maximal ideal of R0. It follows that p
p
n>0
m
H (R= n) = 0 for n > 0. If dim R= > 0, we may choose a homogeneous element x R= , x = 0, of degree 1. Here we use the fact that R is homogeneous. The m
2
p
6
p
p
exact sequence
0 ;! (R= )(;1) ;! R= ;! R=(x ) ;! 0 gives the equation H (R= n) = H (R= n + 1) ; H (R= n) = H (R=(x ) n + 1): As dim R=(x ) = d ; 1, our induction hypothesis implies that H (R= n) is of polynomial type of degree d ; 2. Hence if d > 1, then 4.1.2 implies that d ;1H (R= n) = d ;2(H (R= n)) is a non-zero constant function for n, and if d = 1, then d ;1H (R= n) = H (R= n) = H (R= 0) + Pn large i=1 H (R=( x) i) is constant for large n since H (R=( x) i) = 0 for i 0. Again this constant is not zero since H (R= 0) 6= 0. Now 4.1.2, (a) ) (b) yields the assertion. Hilbert's original proof (see 171]) makes use of his syzygy theorem 2.2.14. This approach will be described in 4.1.13. The next lemma clari es which polynomials in QX ] have integer values. Lemma 4.1.4. Let P (X ) 2 QX ] be a polynomial of degree d ; 1. Then x
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
the following conditions are equivalent: (a) P (n) Z for all n Z (b) there exist integers a0 . . . ad ;1 such that
2
2
P (X ) =
X X + i d ;1 i=0
ai
i
:
is trivial. For the converse observe that the P polynomials ; 2 N), (a) d ;1 ;X +i form a Q -basis of Q X ]. Therefore P ( X ) = i=0 ai i i ; ; ;
Proof. (b) X +i ,i
with ai 2 Q. The identity X +ii+1 ; Xi+i = Xi;+1i immediately implies that ai = i P (;i ; 1) 2 Z for i = 0 . . . d ; 1. De nition 4.1.5. Let M be a nite graded R -module of dimension d . The unique polynomial PM (X ) 2 QX ] for which H (M n) = PM (n) for all n 0 is called the Hilbert polynomial of M . We write PM (X ) =
X
d ;1 i=0
(;1)d ;1;ied ;1;i
X + i i
:
150
4. Hilbert functions and multiplicities
Then the multiplicity of M is de ned to be e d > 0, e(M ) = `0(M ) if if d = 0.
Remark 4.1.6. The higher iterated Hilbert functions Hi (M n), i 2 N, of a nite graded R -module M are de ned recursively as follows: X H0 (M n) = H (M n) and Hi (M n) = Hi;1 (M j ) j n
for i > 0. Occasionally the functions Hi (M ) are called the sum transforms of H (M ). It follows from 4.1.2 and 4.1.3 that Hi (M n) is of polynomial type of degree d + i ; 1, where d = dim P M. ;In particular, for all n 0 there is a representation H1 (M n) = di=0 ai n+i i with ai 2 Z, and it is easy to see that ad = e(M ). Another formula for the multiplicity will be given in 4.1.9. Theorem 4.1.3 together with the next lemma yields a structural result about Hilbert series. P Lemma 4.1.7. Let H (t) = antn be a formal Laurent series with integer coecients, and ai = 0 for i 0. Further, let d > 0 be an integer. Then the following conditions are equivalent: (a) there exists a polynomial P (X ) QX ] of degree d 1 such that P (n) = an for large n (b) H (t) = Q(t)=(1 t)d where Q(t) Zt t;1] and Q(1) = 0. Proof. Assume (a), and set F (n) = an for all n Z. Then
2 2
;
6
;
2 (1 ; t)d H (t) = d F (n ; d )tn n and it follows from 4.1.2 that (1 ; t)d H (t) 2 Zt t;1] . We set Q(t) = P d n n F (n ; d )t . Suppose Q(1) = 0 then X X 0 = d F (n ; d ) = (d ;1F (n + 1 ; d ) ; d ;1F (n ; d )) = d ;1F (m) X
n
n
for large m. This contradicts 4.1.2, and thus proves the implication (a) ) (b). The converse is proved similarly. Corollary 4.1.8. Let M 6= 0 be a nite graded R-module of dimension d. Then there exists a unique QM (t) 2 Zt t;1] with QM (1) 6= 0 such that Q (t) HM (t) = M d : (1 ; t) P Moreover, if QM (t) = i hi ti , then minfi : hi 6= 0g is the least number such that Mi 6= 0.
151
4.1. Hilbert functions over homogeneous rings
Proof. The rst part of the assertion is clear for d = 0, and for d > 0 it follows from 4.1.3 and 4.1.7. In order to prove the second part multiply both sides of HM (t) = QM (t)=(1 ; t)d by (1 ; t)d and compare coecients.
In the next proposition we show how one can recover the coecients
ei of the Hilbert polynomial of a module M from QM . We will denote by P (i) the i-th formal derivative of an element P Zt t;1]. Proposition 4.1.9. Under the assumptions of 4.1.8 the following formulas hold: Q(i) (1) ei = M i! for i = 0 . . . d 1. Moreover, e(M ) = QM (1).
2
;
Proof.
We write
HM (t)
; X (;i!1) (1Q;Mt(1))d;i = (1D;(tt))d d ;1
i
(i)
i=0
P where D(t) = QM (t); d ;1 (;1) Q(i) (1)(1;t)i is the remainder of the Taylor i=0
i
expansion of QM (t) up to degree d ; 1. The element D(t) 2 Zt t;1] is (j )(1) = 0 for j = 0 . . . d ; 1. It follows that the divisible by (1 ; t)d since DP ;1( (;1)i Q(i) (1)=(1 ; t)d ;i) coincide for large coecients of HM (t) and di=0 M i! n. Hence d ;1 X (;1)i Q(Mi) (1) = X P (n)tn M i! (1 ; t)d ;i i=0
i!
M
n 0
since the coecients of both series are polynomial functions in n which are equal for large n (and hence must be equal for all n). Expanding the left hand side of the equation as a power series, and comparing coecients we get ei = Q(Mi) (1)=i!. Finally, by what we have just proved, P we have e(M) = e0 = QM (1) if d > 0, and, if d = 0, e(M ) = `(M ) = n H (M n) = HM (1) = QM (1), since in this case HM (t) = QM (t). Corollary 4.1.10. Assume that in addition to P the assumptions of 4.1.8 the module M is Cohen{Macaulay. Let QM (t) = hi ti . Then hi 0 for all i. Moreover, ei 0 for all i if Mj = 0 for all j < 0. Proof. Without loss of generality we may assume that the residue class eld of R0 is in nite. Otherwise we resort to a standard trick: we replace R by R 0 = R R R0(Y ) and M by M 0 = M R R0(Y ) where Y is an indeterminate over R0 , and where R0(Y ) is the local ring R0Y ]S , S being the multiplicatively closed set of polynomials P (Y ) 2 R0Y ] which have at least one unit among their coecients. The natural ring 0
0
152
4. Hilbert functions and multiplicities
homomorphism R0 ! R0(Y ) is local and at, and its bre is the residue class eld of R0(Y ), namely the eld k(Y ) of rational functions over k = R0= 0 . If we assign the degree 0 to the elements of R0 (Y ), then both R 0 and M 0 are naturally graded, and because of atness, M 0 is a Cohen{Macaulay R 0-module of dimension d with HM 0 (t) = HM (t) see 2.1.7 and 1.2.25. Moreover, R00 = R0(Y ), and hence has an in nite residue class eld. In view of 4.1.9 it suces to show that hi 0 for all i. We prove the assertion by induction on d . If d = 0, then QM (t) = HM (t) (see 4.1.8), and so all coecients of QM (t) are non-negative. Suppose L now that d > 0. The unique homogeneous maximal ideal =L 0 n>0 Rn of R does not belong to Ass M , and the ideal I = n>0 Rn, generated by the elements of R1 , is -primary. Thus, since R= is in nite there exists an element a 2 R1 which is M -regular see 1.5.12. Let N = M=aM then N is a Cohen{Macaulay graded R -module of dimension d ; 1, and the exact sequence m
M
m
M
M
0 ;! M (;1) ;! M ;! N ;! 0 gives the equation (1 ; t)HM (t) = HN (t). It follows that QM (t) = QN (t), which by our induction hypothesis yields the conclusion. Remark 4.1.11. The arguments in the previous proof show the following notable result: suppose M is a nite graded R -module, and x is an M -sequence of elements of degree 1 then QM (t) = QM=(x)M (t). Hilbert's theorem tells us that the Hilbert function of a nite graded module is a polynomial function for large n. We will determine from which integer n onwards this happens. Proposition 4.1.12. P Let M 6= 0 be a nite graded R-module of dimension d, and QM (t) = bi=a hiti with hb 6= 0. Then H (M b ; d ) 6= PM (b ; d ) and H (M i) = PM (i) for all i b ; d + 1. ; Proof. For i = a . . . b we set Hi (t) = hi ti =(1 ; t)d and Pi(n) = hi n;di+;d1;1 . P Then Hi (t) = 1n=i Pi(nP )tn, but since Pi (n) = 0 for n = i ; (d ; 1)P . . . i ; 1 we even have Hi (t) = 1n=i;(d ;1) Pi(n)tn. Furthermore PM (n) = bi=a Pi (n) P for all n 2 Z. For n b ; d + 1 one has H (M n) = bi=a Pi (n), whereas P H (M b ; d ) = bi=;a1 Pi(b ; d ) and Pb(b ; d ) 6= 0. Thus H (M b ; d ) 6= PM (b ; d ). In Section 4.4 we will give a homological interpretation of the di erence between the Hilbert function and the Hilbert polynomial. a
Hilbert series and free resolutions. The Hilbert series of a graded module
can be expressed in terms of its graded resolution.
153
4.1. Hilbert functions over homogeneous rings
Lemma 4.1.13. Let M be a nite graded R-module of nite projective dimension, and let
0 ;!
M j
; ;! ;! M R(;j) ;! M ;! 0
R ( j )pj
0j
j
be a graded free resolution of M. Then HM (t) = SM (t)HR (t)
P where SM (t) = ij (;1)i ij tj .
In particular, if R = kX1 . . . Xn] is the polynomial ring over the eld k, then S (t) HM (t) = M n : (1 t)
;
For the proof we simply note that thePHilbert function is additive on short exact sequences, so that HM (t) = i (;1)iij HR(;j ) (t). Taking into account that HR(;j ) (t) = tj HR (t), we obtain the required formula. If R = kX1 . . . Xn] is the polynomial ring, then H (R i) equals the number of monomials; in degree i. One easily Pproves ; by induction on n that this number is nn+;i;1 1 , whence HR (t) = i nn+;i;1 1 ti = 1=(1 ; t)n. Proof.
Corollary 4.1.14. Let R = kX1 . . . Xn] be a polynomial ring over a eld k, and let M be a nite graded R-module of dimension d. Then (a) SM (t) = (1 t)n;d QM (t), (b) n d = inf i : SM(i)(1) = 0 , ; (c) SM(n;d +i)(1) = ( 1)n;d n;di +i ei .
;
Proof.
; f
;
6 g
(a) and (b) are immediate while (c) follows from (a) and 4.1.9.
We conclude this section with an application to a special class of graded rings. Let R = kX1 . . . Xn] be a polynomial ring over a eld k, I R a graded ideal. We say that R=I has a pure resolution of type (d1 . . . dp) if its minimal resolution has the form 0 ;! R (;dp )p ;! ;! R (;d1 ) ;! R ;! R=I ;! 0: Note that d1 < d2 < < dp. Theorem 4.1.15. Suppose R=I is Cohen{Macaulay and has a pure resolution of type (d1 . . . dp ). Then 1
(a) i = (;1)i+1
Y j 6=i
dj
(dj ; di )
Yp (b) e(R=I ) = p1! di : i=1
154
4. Hilbert functions and multiplicities
As R=I is Cohen{Macaulay, the Auslander{Buchsbaum formula 1.3.3 (in conjunction with 1.5.15) implies that p = dim R ; dim R=I therefore, with 0 = 1, d0 = 0, we have Proof.
SR=I (t) =
Xp i=0
(j ) (1) = 0 (;1)i i tdi and SR=I
for j = 0 . . . p ; 1 see 4.1.14. We obtain the following system of linear equations:
Xp i=1
Xp
(;1)ii = ;1
i=1
(;1)ii di (di ; 1) (di ; j + 1) = 0
for j = 1 . . . p ; 1:
Upon applying elementary row operations, which do not a ect the solution of this system of linear equations with coecient matrix (di!=(di ; j )!) j =0...p;1 i=1...p
we are led to the Vandermonde matrix whose determinant is Now Cramer's rule gives the stated solutions for the i. (b) According to 4.1.14 we have
;
e(R=I ) = ( 1)p
Q
i>j
(di ; dj ).
(p) (1) X p SR=I p+i di : = ( 1) i p! p i=0
;
Thus (a) implies that
Q
p Xp p;1(di ; j) 1Y e(R=I ) = di Q j =1 : p! i=1 i=1 j 6=i(di ; dj )
It remains to show that the sum in this expression equals 1. We introduce the rational complex function
Qp;1(z ; j) f (z ) = Qpj =1 (z ; d ) : j =1
j
This function has simple poles at worst in the points d1 . . . dp , and the residues in these points are
Qp;1(d ; j) Resd f (z ) = Q j =1 i : (di ; dj ) i
j 6=i
155
4.1. Hilbert functions over homogeneous rings
The sum of all residues of a rational function at all points including 1 is zero, and Res1 f (z ) = ; Res0 f (1=z )=z 2. Therefore p;1 Xp Y
Y
i=1
j 6=i
(di ; j )
j =1
(di ;
dj );1
! Xp =
i=1
Y = Res0 f (1z 2=z ) = Res0 z1 (1 j =1 p;1
Resdi f (z )
; jz)
Yp
(1 ;
j =1
dj z );1
!
= 1:
Exercises Let k be a eld, and M a nite graded module over the polynomial ring R = k X1 . . . Xn ] with minimal graded resolution 4.1.16.
0 ;;!
M j
;
R ( j )pj
;;! ;;! M R(;j) ;;! M ;;! 0: 0j
j
We say that two modules have numerically the same resolution if their graded Betti numbers ij are the same. Show: (a) The homogeneous rings k X Y ]=(X 2 Y 2 ) and k X Y ]=(X 2 XY Y 3 ) have the same Hilbert series, but their minimal graded free k X Y ]-resolutions are numerically di erent. (b) The homogeneous rings k X Y ]=(X 2 Y 2 ) and k X Y ]=(XY X 2 ; Y 2 ) have numerically the same graded k X Y ]-resolution, but are not isomorphic when k = R. 4.1.17. Let k be a eld, and let R = k X1 . . . Xn ]=I be a homogeneous Cohen{ Macaulay ring. The ring R has an m-linear resolution if it has a pure resolution of type (m m + 1 . . . m + p ; 1), p = n ; dim R . (a) Suppose R has an m-linear resolution. What are the ranks of the free modules in the free resolution of R , and what is the multiplicity of R ? (b) Suppose dim R = 0 prove R has an m-linear resolution if and only if I = (X1 . . . Xn )m . Hint: relate the last shifts in the resolution of R with the degrees of the socle elements of R . (c) Prove the homogeneous Cohen{Macaulay ring R = k X1 . . . ;Xn ]=I has an m-linear resolution if and only if Ij = 0 for j < m, and dimk Im = m+mg 1 , where g = height I . Hint: reduce to dimension zero. 4.1.18. Let k be a eld, and let R = k X1 . . . Xn ]=I be a homogeneous Gorenstein ring of dimension 0. Assume that all generators of I have the same degree c. (a) Show a(R ) 2c ; 2. (b) Show a(R ) = 2c ; 2 if and only if R has a pure resolution of type (c c + 1 . . . c + n ; 2 2c + n ; 2). In this case R is called an extremal Gorenstein ring. This class of Gorenstein rings was rst considered by Schenzel 327]. (c) Compute the Betti numbers i (R ) of an extremal Gorenstein ring R in a minimal graded free k X1 . . . Xn ]-resolution of R . ;
156
4. Hilbert functions and multiplicities
4.2 Macaulay's theorem on Hilbert functions This section is devoted to a theorem of Macaulay describing exactly those numerical functions which occur as the Hilbert function H (R n) of a homogeneous k-algebra R , k a eld. Macaulay's theorem says that for each n there is an upper bound for H (R n + 1) in terms of H (R n), and this bound is sharp in the sense that any numerical function satisfying it can indeed be realized as the Hilbert function of a suitable homogeneous k-algebra. One part of the proof of Macaulay's theorem will be based on a theorem of Green which relates the Hilbert function of a homogeneous ring R with the Hilbert function of the factor ring R=hR by a general linear form h.L Let R = n 0 Rn be a homogeneous k-algebra, where R0 = k is a eld. We will show that R has a k-basis consisting of monomials in a basis x1 . . . xm of R1 . We are going to de ne this basis of monomials on the level of the polynomial ring. So let : kX1 . . . Xm ]
;! R
be the surjective k-algebra homomorphism with (Xi) = xi . De nition 4.2.1. A non-empty set of monomials in the indeterminates X1 . . . Xm is called an order ideal of monomials if the following holds: whenever m 2 and a monomial m0 divides m, then m0 2 . Equivalently, if X1a Xmam 2 and 0 bi ai for i = 1 . . . m, then X1b Xmbm 2 . Remarks 4.2.2. (a) In Chapter 9 we introduce the order ideal of an element in a module. This notion has nothing to do with the order ideal of monomials, and they should not be confused. (b) Of course an order ideal of monomials is not a k-basis of an ideal, let alone an ideal. Quite the contrary, if we let C be the complement of in the set of all monomials, then C is a k-basis of the ideal generated by the monomials m 2 C . Theorem 4.2.3 (Macaulay). Let R be a homogeneous k-algebra, k a eld. Further let x1 . . . xm be a k-basis of R1, and : kX1 . . . Xm ] ! R the k-algebra homomorphism with (Xi) = xi for i = 1 . . . m. Then there exists an order ideal of monomials such that ( ) is a k-basis of R. Proof. Let S denote the set of all monomials in the indeterminates X1 . . . Xm. We de ne a total order, the so-called reverse degree-lexicographical order, on S: if u = X1a Xmam and v = X1b XPmbm , then Pu < v if the last non-zero component of (b1 ; a1 . . . bm ; am , bi ; ai ) is positive. (The usage of the term `reverse degree-lexicographical' is not coherent in the literature.) M
M
M
1
M
1
M
M
M
M
M
M
M
M
1
1
157
4.2. Macaulay's theorem on Hilbert functions
It is clear that (S v2 > of elements of S must stop after a nite number of steps. Equivalently, every non-empty set of elements in S has a minimal element { a fact that will be used later. Now we de ne recursively a sequence of monomials u1 u2 . . . according to the following rule. We set u1 = 1 assume u1 . . . ui have been de ned then we let ui+1 be the least element in the reverse degreelexicographical order such that (u1 ) . . . (ui ), (ui+1) are linearly independent over k. If such ui+1 does not exist, the sequence terminates with ui . We claim that = fu1 u2 . . .g is the required order ideal of monomials. By construction ( ) is a k-basis of R . Assume is not an order ideal of monomials. Then there exist ui 2 and u 2 S nP such that ui = u Xj for some Xj . As u 2= , we can P write (u) = i (ui) with ui 2 , ui < u, and i 2 k. Then (ui ) = i (ui Xj ), and ui Xj < ui , for all i in the sum, a contradiction. We saw in the proof that our k-basis (u1) (uP2 ) . . . of R has a remarkable property: let u 2 S, and write (u) = i6=0 i (ui ). Then ui u for all i, and if u 2= , these inequalities are strict. The previous theorem and 4.2.2(b) immediately imply Corollary 4.2.4. Let J be the ideal which is generated by the monomials in C . Then the homogeneous k-algebra R and kX1 . . . Xm ]=J have the M
M
M
0
0
M
M
M
0
M
0
M
M
same Hilbert function. In particular, all Hilbert functions of homogeneous rings arise as Hilbert functions of homogeneous rings whose de ning ideal is generated by monomials. The set of monomials C associated with R can be described di erently. Let I = Ker , and set L(I ) = L(f ): f I and I ? = L(I )R where L(f ) denotes the leading monomial of f , that is, the monomial occurring in f which is maximal in the reverse degree-lexicographical order.P Then L(I ) = C . Indeed, let v L(I ), and choose f I , f = ni=1 i vi with monomials vi such that v = L(f ) = vn . Assume vn = C then vn , and so M
f
2g
2
M
2
M
2
2
M
0 6= (vn ) = ;
n;1 X
;n 1i (vi ):
i=1 P Each (vi ) is a linear combination ij (uj ), ij 2 k, uj 2
, uj vi < vn . Replacing the (vi ) in the above equation by their linear combinations gives a representation as a non-trivial linear combination of elements in ( ). This contradicts 4.2.3. M
M
158
4. Hilbert functions and multiplicities
P
Conversely, suppose v 2 C P. Then (v) = i (ui ) with ui 2 , ui < v. Hence, if we set f = v ; iui , then (f ) = 0 and L(f ) = v. The ideal I ? is nitely generated. Therefore there exist polynomials f1 . . . fn 2 I such that I ? = (L(f1) . . . L(fn)). Any such subset of I is called a Grobner or standard basis of I . NotePthat any Grobner basis of I generates I : let f 2 I then L(f ) =P giL(fi) for some P gi 2 kX1 . . . Xm], and it follows that either f = gi fi, or L(f ; gi fi ) < L(f ) for a suitable 2 k. In the rst case, f is an element of (fP1 . . . fn ). In the second case we apply the same procedure to f 0 = f ; gifi to obtain an element f 00 which is either zero, in which case f 2 (f1 . . . fn ), or which has L(f 00) < L(f 0 ). Since any descending sequence of elements in S terminates, we eventually arrive at the required conclusion. We should warn the reader that the converse is not true. Consider for example the ideal I = (f1 f2) with f1 = X1X2 + X32, f2 = X2X3. Then X1X22 = X2f1 ; X3f2 is an element of I ? , but not of (L(f1) L(f2)) = (X32 X2X3). Even though a Grobner basis of an ideal I is not simply given by the leading forms of a system of generators of I , there does exist an algorithm to compute a Grobner basis { the so-called Buchberger algorithm. This, and the fact that most explicit calculations in commutative algebra are performed using Grobner bases, explain their importance. Buchberger's algorithm has been implemented in various computer algebra programs. M
M
Macaulay representations and lexsegment ideals. Let S = kX1 . . . Xm] denote the polynomial ring over a eld k. The problem of determining the Hilbert function of a homogeneous factor ring R of S boils down to the following question: given a subspace V Sd , what can be said about the k-dimension of the subspace S1 V in Sd +1? We will give the answer in a special but important case. Let u Sd
2
be a monomial. We de ne the sets Lu = fv 2 Sd : v < ug and Ru = fv 2 Sd : v ug of the monomials of degree d which are `left' and `right' of u. Note that RX = fX1 . . . Xng. Monomial sets of the form Ru Sd are called lexsegments (of degree d ). The next lemma says that a lexsegment of degree d spans a lexsegment of degree d + 1. Lemma 4.2.5. RX Ru = RX u. Proof. Let v 2 Ru then Xi v X1v X1u. Conversely, let v 2 RX u. We may assume that X1 does not divide v then v > X1u. Let u = X1a Xmam , v = X2b Xmbm , and i be the largest integer such that bi > ai . If there exists j < i with bj > 0, then Xj;1v 2 Ru otherwise, Xi;1v 2 Ru . In both cases it follows that v 2 RX Ru. 1
1
1
1
2
1
1
159
4.2. Macaulay's theorem on Hilbert functions
The sets Lu admit a natural decomposition: let i be the largest integer such that Xi divides u. Then we can write Lu = L0u L00u Xi where Xi does not divide any element in L0u. It is clear that this union is disjoint, that L0u consists of all monomials of degree d in the variables X1 . . . Xi;1, and that L00u = LXi; u . An example illustrates this decomposition. Let S = kX1 . . . X4] and u = X22X3. Then Lu = fX13 X12X2 X1X22 X23 X12X3 X1X2X3g L0u = fX13 X12X2 X1X22 X23g L00u = fX12 X1X2g = LX : It is convenient to denote the set of all monomials of degree d in the variables X1 . . . Xi by X1 . . . Xi]d . We may again decompose L00u , etc. Thus if we write u = Xj (1)Xj (2) Xj (d ) with 1 j (1) j (2) j (d ), then Lu = X1 . . . Xj(d);1]d LXj;d uXj(d) = X1 . . . xj (d );1]d X1 . . . Xj (d ;1);1]d ;1Xj (d ) 1
2 2
1 ( )
= and we end up with the disjoint union
LX
;1
;1
j (d ;1) Xj (d ) u
Xj (d ;1)Xj (d )
Lu = X1 . . . Xj(i);1]iXj(i+1) Xj(d) i=1 called the natural decomposition of Lu. d
It follows that
jLuj =
Xd k(i) i=1
i
with k(i) = j (i) + i ; 2. Note that ; k(d ) > k(d ; 1) > > k(1) 0. Here and in the sequel we use that kl = 0 for 0 k < l . The above considerations show that any non-negative integer has such a binomial sum expansion. We prove this directly. Lemma 4.2.6. Let d be a positive integer. Any a 2 N can be written
uniquely in the form
k(d ) k(d ; 1) k(1) a= + + +
where k(d ) > k(d
;
d 1) >
;1 > k(1) 0. d
1
160
4. Hilbert functions and multiplicities
Proof. ; In order to prove; the existence, Pwe choose ; k(d ) maximal such that k(dd ) a. If a = k(dd ) , then a = di=1; k(ii) with k(i) = i ; 1 for i = 1 . . . d ; 1. Now assume that a0 = a ; k(dd ) > 0. By the induction P ;1 ;k(i) with k(d ; 1) > k(d ; 2) > hypothesis we may assume that a0 = di=1 i > k(1) 0. It remains to show that k(d ) > k(d ; 1): since ;k(dd)+1 > a, it follows that k(d ) k(d ) + 1 k(d ) k(d ; 1) = ; > a0 : d ;1 d d d;1 Hence k(d ) > k(d ; 1). The uniquenessPfollows by induction on a, once we have shown the d ;k(i) following: if a = i=1 i with; k(d ) > k(d ; 1) > > k(1) 0, then k(d ) is the largest integer with k(dd ) a. Again we prove this statement by induction ; on a. For a = 1 the assertion is trivial. Now assume that a > 1, and k(dd)+1 a. Then
X k(i) k(d ) + 1 k(d ) k(d ) k(d ; 1) + 1 ; = d ;1 i=1
i
d
d
d
;1
d
;1
and this contradicts the induction hypothesis. Following Green 138] we refer to the sum 4.2.6 as the d-th Macaulay representation of a, and call k(d ) . . . k(1) the d-th Macaulay coecients of a. Note that for all i d the coecient k(i) is determined by the property of being the maximal integer j such that j k(d ) k(i + 1) a; ; ; : i d i+1 The d -th Macaulay coecients have the following nice property. Lemma 4.2.7. Let k(d ) . . . k(1), respectively k0 (d ) . . . k0 (1), be the d-th Macaulay coecients of a, respectively a0. Then a > a0 if and only if
(k(d ) . . . k(1)) > (k0 (d ) . . . k0 (1)) in the lexicographical order.
We prove both implications by induction on d . For d = 1 the assertion is trivial. We now assume that d > 1. If k(d ) = k0 (d ), then k(d ; 1) . . . k(1) (respectively k0 (d ; 1) . . . k0 (1)); are the (d ; 1)-th ; 0 k(d ) Macaulay coecients of a ; d (respectively a0 ; k d(d ) ), and we may apply the induction hypothesis. If k(d ) 6= k0 (d ), then k(d ) > k0 (d ) if
Proof.
161
4.2. Macaulay's theorem on Hilbert functions
and only if a > a0 . This follows from the characterization of the d -th Macaulay coecients preceding this lemma. Skipping the summands which are zero in the d -th Macaulay representation of a we get the following unique sum expansion: k(d ) k(d ; 1) k(j) a= + d ; 1 + + j d where k(d ) > k(d ; 1) > > k(j ) j 1. We de ne k(d ) + 1 k(d ; 1) + 1 ahd i = + + + k(1) + 1 k(dd +) +1 1 k(d ;d1) + 1 k(j)2+ 1 = + + + d+1 d j+1 and set 0hd i = 0. Proposition 4.2.8. Let u be a monomial of degree d in the polynomial ring S. Then jLX uj = jLujhd i . 1
S
Let Lu = di=1X1 . . . Xj (i);1]i Xj (i+1) Xj (d ) be the canonical decomposition of Lu. We claim that Proof.
d
X1 . . . Xj (i);1]i+1Xj (i+1) Xj (d )
i=1
is the canonical decomposition of LX u. Indeed, the canonical decomposition of Lu is completely determined by the sequence j (1) . . . j (d ) attached to u. Let l (1) . . . l (d +1) be the corresponding sequence for X1u. Then l (1) = 1, and l (i) = j (i ; 1) for i = 2 . . . d + 1. This proves the claim and the proposition. Let S = kX1 . . . Xm ] be a polynomial ring over a eld, and u 2 S a monomial. An ideal which in each degree is spanned by a lexsegment will be called a lexsegment ideal. In view of 4.2.5 and 4.2.8 we obtain Corollary 4.2.9. Let I S be a lexsegment ideal, and set R = S=I. Then 1
H (R n + 1)
H (R n)hni
for all
n:
Equality holds for a given n if and only if In+1 = (X1 . . . Xm)In . Macaulay's theorem. We now come to the main result of this section. It
will follow that the growth of the Hilbert function of a homogeneous ring de ned by a lexsegment ideal is, in a sense, the maximum possible.
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4. Hilbert functions and multiplicities
Theorem 4.2.10 (Macaulay). Let k be a eld, and let h : N ! N be a
numerical function. The following conditions are equivalent: (a) there exists a homogeneous k-algebra R with Hilbert function H (R n) = h(n) for all n 0 (b) there exists a homogeneous k-algebra R with monomial relations and with Hilbert function H (R n) = h(n) for all n 0 (c) one has h(0) = 1, and h(n + 1) h(n)hni for all n 1 (d) let m = h(1), and for each n 0 let n be the rst h(n) monomials in the variables X1S . . . Xm of degree n in the reverse degree-lexicographical order set = n 0 n then is an order ideal of monomials.
M
M M
M
The following example demonstrates the e ectiveness of Macaulay's theorem: let us check that 1 + 3t + 5t2 + 8t3 is not the Hilbert series ;3 + of;2a, homogeneous ring. In fact, condition (c) is violated since 5 = 2 1 ; ; and 5h2i = 43 + 32 = 7 < 8. Instead we also could apply (d), and get M1 = fX1 X2 X3g, M2 = fX12 X1X2 X22 X1X3 X2X3g, M3 = fX13 2X12X2 X1X22 ;X1 23 X122X3 X1X2X3 X22X3 X1X32g. Thus we see that X1X3 2 M3 , but X1 (X1X3 ) 2= M2. Therefore M is not an order ideal of monomials. Most parts of the theorem have already been shown: the equivalence of (a) and (b) is the content of 4.2.4, and the implication (d) ) (b) is trivial. For the proof of (c) ); (d) we assume that h(1) = m. ;Then condition (c);implies that h(n) n+mn ;1 . Suppose that h(n + 1) = nn++1m , then h(n) = n+mn ;1 , and so Mi = X1 . . . Xm]i for i = n and i = n + 1. Therefore, if u 2 Mn+1, and Xi divides u, then trivially Xi;1u 2 Mn . ; n+m Now we suppose that h(n + 1) < n+1 , then there exist a monomial un+1 such that Mn+1 = Lun . If, as before, Mn = X1 . . . Xm ]n, there is nothing to show. Otherwise, there exists a monomial un such that Mn = Lun . Condition (c) and 4.2.8 imply that RX Run Run . Therefore, if u 2S Mn+1, and Xi divides u, then Xi;1u 2 Mn . In other words, M = n 0 Mn is an order ideal of monomials. For the most dicult implication (a) ) (c) we present the elegant proof of Green 138]. This needs some preparations. If a positive integer a has d -th Macaulay coecients k(d ) . . . k(1), then let k(d ) ; 1 k(d ; 1) ; 1 k(1) ; 1 ahd i = + + + d;1 1 k(d )d; 1 k(j) ; 1 = + + +1
1
+1
d j where j = min i : k(i) i . Note that ahd i has d -th Macaulay coecients k(d ) 1 . . . k(j ) 1 j 2 j 3 . . . 0.
;
f
g
;
; ;
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4.2. Macaulay's theorem on Hilbert functions
Lemma 4.2.11. (a) If a a0 , then ahd i a0hd i . (b) If k(j ) 6= j for j = minfi : k(i) ig, then (a ; 1)hd i < ahd i . Proof. (a) follows from the observation preceding this lemma and 4.2.7. For (b) let k(d ) k(d ; 1) . . . k(1) be the d -th Macaulay coecients of a, and k0 (d ) k0 (d ; 1) . . . k0 (1) the d -th Macaulay coecients ; ofa ; 1 then k0 (d ) k(d ) by 4.2.7. If k0 (d ) = k(d ), we set a0 = a ; k(dd ) . Convince yourself that a0 ; 1 > 0. Then it follows that a0 (respectively a0 ; 1) has (d ; 1)-th Macaulay coecients k(d ; 1) k(d ; 2) . . . k(1) (respectively k0 (d ; 1) k0 (d ; 2) . . . k0 (1)). Moreover, a0 satis es the hypothesis of (b). Therefore, if we argue by induction on d , we may assume that (a0 ; 1)hd ;1i < a0hd ;1i . Hence the required inequality in the case in ; which k0 (d ) = k(d ) follows from the equalities ahd i = a0hd ;1i + k(dd);1 and ; (a ; 1)hd i = (a0 ; 1)hd ;1i + k(dd);1 . Now suppose that k0 (d ) < k(d ). Our assumption implies that the d -th Macaulay coecient of ahd i is k(d ) ; 1, and that the d -th Macaulay coecient of (a ; 1)hd i is less than or equal to k0 (d ) ; 1. Therefore the conclusion follows from 4.2.7. Theorem 4.2.12, interesting in its own right, is the key to the still unproved implication (a) ) (c) of 4.2.10. Let R be a homogeneous k-algebra, k an in nite eld. The ane k-space R1 is irreducible, and so any non-empty (Zariski-) open subset is dense in R1. This suggests the following terminology: a property P holds for a general linear form of R1 if there exists a non-empty open subset U of R1 such that P holds for all h 2 U . Theorem 4.2.12 (Green). Let R be a homogeneous k-algebra, k an in nite eld, and let n 1 be an integer. Then H (R=hR n) H (R n)hni for a general linear form h. Proof. Let s = sup dimk hRn;1 : h R1 then dimk hRn;1 = s for a general linear form. Indeed, let U R1 be the subset of elements h R1 such that dimk hRn;1 = s. It is obvious that U = . In order to see that U is open, we choose a basis a1 . . . am of R1 and bases P of Rn;1 and Rn. Then the multiplication map h : Rn;1 Rn, h = mi=1 xi ai, can be described by a matrix of linear forms in x1 . . . xm . Replacing the xi by indeterminates yields a matrix A of linear polynomials with coecients in k, and it is clear that U is the complement of V (Is (A)) in R1. Let h U , and set S = R=hR . We claim that H (S n) H (R n)hni , and prove it by induction on min n dimk R1 . If either n = 1 or dimk R1 = 1, then the assertion is trivial. Now assume that n > 1 and dimk R1 > 1. Let V S1 be the subset of linear forms g for which dimk gSn;1 is maximal,
f
2 g
!
2
f
g
2
6
164
4. Hilbert functions and multiplicities
and denote by ' the canonical epimorphism R ! S . We consider the open subset W = (U n kh) \ ';1(V ) of R1. The set W is non-empty since R1 is irreducible and both ';1(V ) and U n kh are non-empty. In fact, assume that U kh. Then, since U is a dense and kh is a closed subset in R1 it follows that R1 = kh, contradicting the assumption dimk R1 > 1. Now we choose h 2 W , and get H (S n) = dimk (Sn =hSn;1) + dimk h Sn;1: By our choice of h, the induction hypothesis yields the inequality dimk (Sn =hSn;1) H (S n)hni :
To obtain an upper bound for the second summand note rst that
dimk hSn;1 dimk (h Rn;1=h(hRn;2)) = dimk (hRn;1=h (hRn;2)): The last equality holds true since the di erence of both sides equals dimk (Rn =hRn;1) ; dimk (Rn=hRn;1), and this di erence is zero since both h and h belong to U . Let W R1 be the (non-empty) open set of linear forms l for which l (hRn;2) has maximal dimension. Then, if we actually choose h 2 W \ W , noting that hRn;1 may be viewed as the (n ; 1)-th homogeneous component of P = R=(Ann h), we may apply our induction hypothesis to conclude that dimk hSn;1 (H (R n) ; H (S n))hn;1i . The rest of the proof is a purely numerical argument. What we need is this: given integers 0 < b < a such that b
bhni + (a ; b)hn;1i
then b ahni . ; ; Assume this fails, and write b = k(nn) + + k(jj ) with k(n) > ; ; j ) j > 0. Then a < k(nn)+1 + + k(jj)+1 , and so a ; b < ;k(n)> +k( ; + jk;(j1) . n;1 ;k(n) + + ;k(2), and We distinguish two cases. If j = 1, then a ; b n;;1 1; ; ; k(n);1 k(1);1 hence (a ; b)hn;1i k(nn;);1 1 + + k(2);1;1 , and bhni = + + n ; 1 . ; k(1);1 Thus our hypothesis implies b k(nn) + + k(2) + < b, a 2 1 contradiction. ; ; If j > 1, then (a ; b)hn;1i < k(nn;);1 1 + + k(jj;);1 1 , and this together with our assumption again yields a contradiction. In order to complete the proof of Macaulay's theorem another numerical result is needed.
165
4.2. Macaulay's theorem on Hilbert functions
Lemma 4.2.13. Let a, a0, and d be positive integers. (a) If a a0 , then ahd i a0hd i . (b) Let k(d ) . . . k(1) be the d-th Macaulay coecients of a, and j = minfi : k(i) ig. Then (a + 1)hd i
=
ahdi + k(1) + 1 ahd i + 1
if j = 1, if j > 1.
Claim (a) follows from 4.2.7, and (b) is immediate for j > 1. Now assume that j = 1, and let i be the maximal integer such that k(i) = k(1) + i ; 1. Then Proof.
a=
=
k(d ) d
+ +
d
+ +
k(d )
and hence a+1 =
k(d ) d
k(i + 1) Xi k(1) + r ; 1 i+1
+
r
k(i + 1) r=1k(1) + i i+1
+ +
+
i
; 1
k(i + 1) k(1) + i i+1
+
i
is the d -th Macaulay expansion of a + 1 since k(i + 1) > k(1) + i. Now we get ahd i =
k(d ) + 1
d+1
Xi k(1) + r + + k(i +i +1)2+ 1 + r+1 r=1
i+1 X k(1) + r ; 1 = k(d ) + 1 + + k(i + 1) + 1 + d+1 i+2 r k(d ) + 1 k(i + 1) + 1 r=2k(1) + i + 1 = d + 1 + + + ; k(1) ; 1 i+2 i+1 and so k(d ) + 1 k(i + 1) + 1 k(1) + i + 1 h d i (a + 1) = d + 1 + + + i+2 i+1 h d i = a + k(1) + 1 as asserted. Proof of 4.2.10, (a) ) (c). We may assume that k is in nite: if necessary replace R by l k R where l is an in nite extension eld of k. Let g be a linear form, and set S = R=gR . The exact sequence 0 ;! gRn ;! Rn+1 ;! Sn+1 ;! 0
166
4. Hilbert functions and multiplicities
yields the inequality H (R n + 1) H (R n) + H (S n + 1). Set a = H (R n) and b = H (R n + 1). For a general linear form g the inequality and 4.2.12 give b a + bhn+1i. Let k(n + 1) . . . k(1) be the (n + 1)-th Macaulay coecients of b. Then k(n + 1) ; 1 k(1) ; 1 bhn+1i = + + n+1 1 and so k(n + 1) ; 1 k(2) ; 1 k(1) ; 1 a + + + : n 1 0 Let, as before, j = minfi : k(i) ig. If j > 1, then k(1) = 0, and k(n + 1) ahni + + k(2) = b: n+1 2 If j = 1, then k(n + 1) k(3) k(2) h n i a + + + + k(2) n+1 3 2 by 4.2.13. But k(2) > k(1), and hence ahni > b. Corollary 4.2.14. Let R be a homogeneous k-algebra, k a eld. Then H (R n + 1) = H (R n)hni for n 0. Proof. Write R = S=I , S = kX1 . . . XS m]. According to 4.2.10(d) there exists an order ideal of monomials M = n 0 Mn in S such that H (R n) = H (S=J n) for all n, where J is the ideal generated by all the monomials not in M. Moreover, the choice of M was such that Mn consists of all monomials of degree n if In = 0 and Mn = Lun for a suitable monomial un otherwise. Since J is nitely generated, there exists an integer r such that Run = RX Run for all n r. Thus the assertion follows from 4.2.8. If we combine 1.5.12 with Macaulay's theorem, we obtain the following characterization of the Hilbert series of Cohen{Macaulay homogeneous algebras. Proposition 4.2.15. Let k be a eld, and h0 . . . hs a nite sequence of +1
1
positive integers. The following conditions are equivalent: (a) there exist an integer d, and a Cohen{Macaulay (reduced) homogeneous k-algebra R of dimension d (whose de ning ideal is generated by squarefree monomials) such that P s h ti HR (t) = i=0 id (1 t)
(b) h0 = 1, and 0 hi+1
;
hhi ii for all i = 1 . . . s
; 1.
167
4.2. Macaulay's theorem on Hilbert functions
(a) ) (b): By 1.5.11 there exists an R -sequence x = x1 . . . xd of degree 1 elements. According to 4.1.8 we have
Proof.
HR (t) =
QR (t) (1 t)d
;
Xs
QR (t) =
i=0
hi ti:
Let R = R=xR then HR (t) = (1 ; t)d HR (t) = QR (t). It follows that H (R n) = hn for all n 0. Therefore 4.2.10 yields the assertion. (b) ) (a): By 4.2.10 there exists a homogeneous k-algebra R = kPX1 . . . Xm]=I , where I is generated by monomials, such that HR (t) = s i i=0 hi t . The k-algebra R is Cohen{Macaulay, simply because R is of dimension zero. In order to get a reduced such k-algebra with the required Hilbert series we consider a certain `deformation' of R as described in the next lemma. Lemma 4.2.16. Let R = kX1 . . . Xm ]=I be a homogeneous k-algebra,
where k is a eld and I is generated by monomials. Then there exist a reduced homogeneous k-algebra S whose de ning ideal is generated by squarefree monomials, and an S-sequence y of elements of degree 1 such that R = S=y S. Proof. Assume I = (u1 . . . un), ui = X1ai Xmaim for i = 1 . . . n. If all exponents aij are at most 1, then I is a radical ideal see Exercise 4.4.17. Suppose now that at least one aij > 1, say ai1 > 1 for some i. We introduce a new indeterminate Y , and set
1
vk = Y ak ;1X1 X2ak 1
2
Xma
km
if ak1 > 1, and vk = uk otherwise. The vi satisfy the following conditions: (i) if Y divides vi , then X1 divides vi (ii) the indeterminate X1 occurs in each vi with multiplicity at most 1. We claim that Y ; X1 is regular modulo the ideal J = (v1 . . . vn ). Indeed, assume the contrary is true. Then there exists an associated prime ideal of kX1 . . . Xm Y ]=J with Y ; X1 2 . By Exercise 4.4.15, is generated by a set of variables, and so X1 Y 2 . It follows that there exists w 2 kX1 . . . Xm Y ], w 2= J , with X1w 2 J and Y w 2 J . As J is generated by monomials, we may assume that w is a monomial. Then there exist integers i, j and monomials u1 , u2 such that X1w = vi u1 and Y w = vj u2. As Y divides vj , it follows from (i) that X1 does also, and so X1 divides w. But then the multiplicity of X1 in vi is at least 2, a contradiction. If all variables in the vi occur with multiplicity one, then J is a radical ideal. Otherwise we repeat this construction, and eventually reach the goal, since at each step we lower the multiplicities of the variables in the generators. p
p
p
p
168
4. Hilbert functions and multiplicities
Exercises Let R be a homogeneous k-algebra, k a eld. (a) Establish from 4.2.14 that there exist integers a1 a2 aj 0 such that n + a1 n + a2 ; 1 n + aj ; (j ; 1) PR (n) = + + + : a a a 4.2.17.
1
2
j
(b) Determine the dimension and the multiplicity of R in terms of the integers a1 . . . aj . 4.2.18. Let k be a eld, S = k X1 . . . Xm ], and u = X1a1 X2a2
Xma
m
= Xj(1) Xj(2) Xj(d )
j (1)
j(2) j(d )
a monomial of degree d . Set R = S=I , where I is the lexsegment ideal generated by Ru . Then deduce (a) dim R = j (d ) ; 1, and (b) e(R ) = ai where i = maxfj : aj 6= 0g, provided dim R > 0.
4.3 Gotzmann's regularity and persistence theorem Gotzmann's 136] regularity and persistence theorems give some deeper insight into the nature of the Hilbert polynomial and the Hilbert function. As before let S = kX1 . . . Xm ] be the polynomial ring in m variables de ned over a eld k, I S a graded ideal, and R = S=I . The regularity theorem is a statement about the regularity of the ideal sheaf I associated with I in projective space. Note that di erent ideals may yield the same ideal sheaf. The ideal I = Ker(S ! R=H 0 (R )) is called the saturation of I the sheafs associated with ideals I and J coincide if and only if I = J . We will formulate Gotzmann's theorems in the language of commutative algebra. So we de ne the (Castelnuovo{Mumford) regularity of a nite graded S -module M , rather than that of a sheaf. It is the number m
reg M = maxfi + j : H i (M )j 6= 0g:
Let q be an integer. Then M is called q-regular if q reg(M ), equivalently, if H i (M )j ;i = 0 for all i and all j > q. Before we set out for Gotzmann's theorems, we include an interesting description of regularity in terms of graded Betti numbers. It shows that reg(M ) measures the `complexity' of the minimal graded free resolution of M . Therefore regularity plays an important r^ole in algorithmic commutative Denoting by M q the truncated graded R -module L Mj , onealgebra. has j q Theorem 4.3.1 (Eisenbud{Goto). The following conditions are equivalent: (a) M is q-regular (b) TorSi (M k)j +i = 0 for all i and all j > q m
m
169
4.3. Gotzmann's regularity and persistence theorem
(c) M q admits a linear S-resolution, i.e., a graded resolution of the form 0 ;! S (;q ; l )cl
Proof. (b)
;! ;! S (;q ; 1)c ;! S (;q)c ;! M q ;! 0: 1
0
() (c): By de nition, the module M q has a linear resolution
if and only if
TorS (M q k)r i
= Hi (x M q )r = 0
for all i r, r 6= i + q:
Here H (x M ) is the Koszul homology of M with respect to the sequence x = X1 . . . Xm . Since (M q )j = 0 for j < q, we always have Hi (x M q )r = 0 for .
r < i + q, while for r > i + q
Hi (x M q )r = Hi (x M )r = TorSi (M k)r :
Thus the desired result follows. (a) ) (c): We may assume q = 0 and M = M 0 . Then it is immediate that H 0 (M ) is concentrated in degree 0. This implies M = H 0 (M ) M=H 0 (M ). The rst summand is a direct summand of copies of k. Hence M is 0-regular if and only if M=H 0 (M ) is 0-regular. In other words, we may assume that depth M > 0. We may further assume that k is in nite. Then there exists an element y 2 S of degree 1 which is M -regular. From the cohomology exact sequence associated with m
m
m
m
0 ;! M (;1) ;! M ;! M=yM ;! 0 y
it follows that M=yM is 0-regular. By induction on the dimension on M , we may suppose that M=yM has a linear S=yS -resolution. But if F is a minimal graded free S -resolution, then F =yF is a minimal graded S=yS -resolution of M=yM . This implies that F is a linear S -resolution of M . (c) ) (a): Again one may assume q = 0 and M = M 0 . Then M has a linear resolution .
.
.
.
;! S (;2)c ;! S (;1)c ;! S c ;! M ;! 0: 2
1
0
Computing ExtiS (M S ) from this resolution we see that ExtiS (M S )j = 0 for j < ;i. By duality (see 3.6.19) there exists an isomorphism of graded R -modules ; H i (M ) = Homk ExtmS ;i (M S (;m)) k : Therefore H i (M )j ;i = 0 for all j > 0, as desired. The regularity theorem says that the regularity of the saturation of an ideal I can be read o the Hilbert polynomial of S=I . m
m
170
4. Hilbert functions and multiplicities
Theorem 4.3.2 (Gotzmann). Write the Hilbert polynomial PR (n) of R = S=I
in the unique form PR (n) =
n + a1 n + a2 ; 1 +
a1
a2
+ +
n + as ; (s ; 1) as
as 0, as described in 4.2.17. Then the saturation I
with a1 a2 of I is s-regular.
We prove the theorem by induction on the dimension of S . For m = 1 the assertion is trivial. Now let m > 1, and choose be a general linear form h. Since PR (n) = PS=I (n) we may assume that I = I . We may further assume that I = S . Then depth R > 0, and h is R -regular. Hence Proof.
6
we get an exact sequence
0 ;! R (;1) ;! R ;! R=hR ;! 0 h
yielding the equation (1)
PR=hR (n) = PR (n)
; PR (n ; 1):
Set Re = R=hR , Se = S=hS then Re = Se=J for some ideal J Se. Furthermore P~R (n) = P~S =J (n). Suppose that (2) P~S =J (n) =
n + b1 n + b2 ; 1 +
b1
b2
+ +
n + br ; (r ; 1)
then J is r-regular by the induction hypothesis. (1) and (2) imply PR (n) =
n + a1 n + a2 ; 1 +
a1
a2
br
+ + n + ar ; (r ; 1) + c ar
where c is a constant and ai = bi + 1 for all i. We claim that c 0, and that I is s-regular for s = r + c. These two claims complete the proof. Indeed, we may set ar+1 = = ar+c = 0. In order to derive the rst claim, assume that c < 0. For n 0 we then have n + a1 n + a2 ; 1 n + ar ; (r ; 1) (3) H (R n) < + + + : a1
a2
ar
The right hand side b of this inequality satis es the equation b=
n + a n + a ; 1 1 2 + + + n + ar ; (r ; 1) n
n
;1
n
; (r ; 1)
171
4.3. Gotzmann's regularity and persistence theorem
so that bhni =
=
n + b1 n + b2 ; 1 +
n +n b1 n +n b;2 1; 1 b1
+
b2
+ + + +
n + br ; (r ; 1) n +n ;br (;r ;(r 1); 1) br
:
Observing that n + ar ; (r ; 1) > n ; (r ; 1), one deduces from (3) (see 4.2.11) that H (R n)hni < bhni . Therefore by Green's theorem, n + b1 n + b2 ; 1 n + br ; (r ; 1) e H (R n) < + + + : b1
b2
br
This contradicts (2). For the proof of the second claim note rst that H i (J ) = H i (J ) for i > 1. Therefore and since J is r-regular we deduce from the local cohomology sequence associated with m
m
0 ;! I (;1) ;! I ;! J ;! 0 that H i (I )j ;i = 0 for all i > 2 and j > r (and thus for j > s). It remains to be shown that H 2 (I )j ;2 = 0 for j > s. Suppose this is not the case, and let j be the largest number with H 2 (I )j ;2 6= 0. Then by 4.4.3(b) below, H (R j ; 2) ; PR (j ; 2) = ;H 1 (R )j ;2 < 0 since H 0 (R ) = 0 and H i;1 (R )j ;2 = H i (I )j ;2 = 0 for i > 2, as we have already seen. By our choice of j we have H 1 (R )j ;1 = 0, so that H (R j ; 2) < PR (j ; 2) but H (R j ; 1) = PR (j ; 1): If j = s + 1, then j ; 2 = s ; 1, and s ; 1 + a1 1 + as;1 + + + 1 PR (s ; 1) = 1 s;1 whence s ; 1 + a1 1 + as;1 H (R j ; 2) = H (R s ; 1) + + : s;1 1 Thus Macaulay's theorem implies s + a1 2 + as;1 h j ;2i H (R j ; 1) H (R j ; 2) s + + 2 = PR (s) ; (as + 1) < PR (s) = PR (j ; 1) which is a contradiction. h
m
m
m
m
m
m
m
m
172
4. Hilbert functions and multiplicities
If j > s + 1, then PR (j ; 2)hj ;2i = PR (j ; 1) (see 4.2.14). We apply Macaulay's theorem again, and get H (R j ; 1) H (R j ; 2)hj ;2i < PR (j ; 1) leading to the same contradiction. By 4.2.14 we have H (R n + 1) = H (R n)hni for all large n. But could it happen that H (R n + 1) = H (R n)hni, and H (R r + 1) < H (R r)hri for some r and n with r > n? The following persistence theorem answers this question. Theorem 4.3.3 (Gotzmann). Suppose that H (R n +1) = H (R n)hni for some n and that I is generated by elements of degree n. Then H (R r + 1) = H (R r)hri for all r n. Proof. We prove the theorem by induction on m = dim S . If m = 1, I is principal, and the assertion is trivial. Now let us assume that m > 1. Let k(n) H (R n) = + + k(1) n 1 be the n-th Macaulay expansion of H (R n). Macaulay's theorem implies r ; n + k(n) r ; n + k(1) (4) H (R r) + + r r;n+1 for all r n, and it remains to be shown that equality holds. Let h be a general linear form. Then (5) (H (R n)hni)hni H (R=hR n)hni H (R=hR n + 1) H (R n + 1) ; H (R n) = (H (R n)hni)hni: The rst inequality is Green's theorem, the second is Macaulay's, the third follows from the exact sequence
; ;!h R ;! R=hR ;! 0
R ( 1)
and the last equality results from the hypothesis that H (R n + 1) =
H (R n)hni.
Since the rst and last term in this chain of inequalities coincide, we must have equality everywhere. In particular, H (R=hR n + 1) = H (R=hR n)hni. Since the de ning ideal of R=hR is again generated by elements of degree n, the induction hypothesis applies and yields H (R=hR r + 1) = H (R=hR r)hri for all r n. One also deduces from (5) that k(n) k(1) H (R=hR n + 1) = (H (R n)hni)hni = + + n+1 2 :
173
4.4. Hilbert functions over graded rings
Therefore PR=hR (r) =
r + (k(n) ; n ; 1) k(n)
;n;1
+ + r + (k(1) ; 2) ; (n ; 1) k(1) ; 2
for all r. Hence the saturation J of the de ning ideal of R=hR is n-regular by the regularity theorem. Let I again denote the saturation of I and set R = S=I . It follows just as in the proof of the regularity theorem that r + (k(n) ; n) r + (k(1) ; 1) ; (n ; 1) (6) PR (r) = + + +c k(n) ; n k(1) ; 1 with c 0. Suppose c > 0 since PR (n) = PR (n), the inequality (4) then implies r ; n + k(n) r ; n + k(1) PR (r) = + + r ; n + 1 + c r H (R r) + c > H (R r) for all r n. This is a contradiction. Now (6) and Gotzmann's regularity theorem entail that I is n-regular, whence H (R r) = PR (r) for all r n. Thus for all r n we obtain the following string of inequalities: H (R r) H (R r) PR (r) = PR (r) = H (R r): Hence equality holds everywhere, and this proves the theorem. Exercise
Let S = k X1 . . . Xm ] be a polynomial ring over a eld k, and let n 1 be an integer. A subspace V of the k-vector space Sn is called a Gotzmann space if the ideal I generated by V satis es H (S=I n + 1) = H (S=I n) n . (a) According to 4.2.9, lexsegments span Gotzmann spaces. Give an example of a set of monomials which is not a lexsegment (even after a permutation of the variables), but spans a Gotzmann space. (b) Let I be the ideal generated by a Gotzmann space V Sn . It can be shown that the ideal I has a linear resolution. Compute the Betti numbers of I . (c) Suppose that dim R=I = 0. Show I is generated by a Gotzmann space if and only if I = n for some n > 0 where = (X1 . . . Xm ).
4.3.4.
h i
m
m
4.4 Hilbert functions over graded rings In this section we considerLpositively graded k-algebras, that is, graded kalgebras of the form R = i 0 Ri where R0 = k and R is nitely generated over k. For simplicity we will assume that k is a eld. In contrast to a homogeneous k-algebra the generators of a positively graded k-algebra may be of arbitrarily high degree. In analogy with 4.1.8 we have
174
4. Hilbert functions and multiplicities
Proposition 4.4.1. Let R be a positively graded k-algebra, k a eld, and M 6= 0 a nite graded R-module of dimension d. Then there exist positive integers a1 . . . ad , and Q(t) 2 Zt t;1] such that Q(t) HM (t) = Qd with Q(1) > 0: ai i=1(1 ; t ) We prove the assertion by induction on the dimension d of M . If d = 0, then dimk M < , and so Mn = 0 for n 0. Therefore HM (t) Zt t;1], and we set Q(t) = HM (t). It is clear that Q(1) = dimk M L > 0. Now assume that d > 0, and let U = H 0 (M ), where = i>0 Ri is the unique graded maximal ideal of R . Note that U is a graded submodule of M with dimk U < , and that = Ass(M=U ). We may assume that k is in nite (see the proof of 4.1.10). Then, according to 1.5.11, there exists a homogeneous (M=U )-regular element x , say of degree a1. The exact sequence Proof.
1
2
m
m
1
m
2
2
m
0 ;! (0 : x)M (;a1) ;! M (;a1 ) ;! M ;! M=xM ;! 0 where (0 : x)M = fu 2 M : xu = 0g, gives rise to the equation HM (t)(1 ; ta ) = HM=xM (t) ; P (t) where P (t) is the Hilbert series of (0 : x)M (;a1 ). The series P (t) actually belongs to Zt t;1] since (0 : x)M U , and U is of nite length. By the induction hypothesis there exist Q (t) 2 Zt t;1], and positive integers a2 . . . ad such that (t) Q (1) > 0: HM=xM (t) = Qd Q ai i=2(1 ; t ) Q Set Q(Qt) = Q (t) ; P (t) di=2(1 ; tai ) then, as required, we have HM (t) = Q(t)= di=1(1 ; tai ) with Q(1) > 0. Remarks 4.4.2. (a) Proposition 4.4.1 is analogously valid in the case where R0 is an Artinian local ring. (b) It can easily be veri ed that the integers a1 . . . ad found in the proof of 4.4.1 are the degrees of elements generating a Noether normalization of R= Ann M . (Also see Exercise 4.4.12.) A function P : Z ! C is called a quasi-polynomial (of period g) if there exist a positive integer g and polynomials Pi , i = 0 . . . g ; 1, such that for all n 2 Z one has P (n) = Pi (n) where n = mg + i with 0 i g ; 1. In the following theorem we consider the graded components of the modules H i (M ). Note that they are nite dimensional k-vector spaces (why?). x
1
m
175
4.4. Hilbert functions over graded rings
Theorem 4.4.3 (Serre). Let R be a positively graded k-algebra, k a eld, and M = 0 a nite graded R-module of dimension d, and denote the maximal ideal of R by . Then (a) there exists a uniquely determined quasi-polynomial PM with H (M n) = PM (n) for all n 0, P (b) H (M n) PM (n) = di=0( 1)i dimk H i (M )n for all n Z, (c) one has
6
m
;
;
2
m
deg HM (t) = maxfn : H (M n) 6= PM (n)g = maxfn :
Xd
(;1)i dimk H i (M )n 6= 0g: m
i=0
(Here deg HM (t) denotes the degree of the rational function HM (t).) Proof. (a) follows from Exercise 4.4.10 or Exercise 4.4.11. (b) holds when d = 0, since then PM = 0 and M = H 0 (M ) whereas H i (M ) = 0 for i > 0. Next one notes that both sides of the equation change by the same amount, namely dimk H 0 (M )n, if one replaces M by M=H 0 (M ). As in the proof of 4.4.1 we may thus assume that H 0 (M ) = 0 and that contains a homogeneous M -regular element x of degree e. Then we have an exact sequence m
m
m
m
m
m
0 ;! M (;e) ;! M ;! M=xM ;! 0: x
Set HM0 (t) =
P
n2Z
(H (M n) ; PM (n))tn and
HM00 (t) =
X Xd
( (;1)i dimk H i (M )n)tn:
n2Z i=0
m
As HM=xM (t) = (1 ; te)HM (t), it follows that PM=xM (n) = PM (n) ; PM (n ; e) for all n 0, and, hence, PM=xM (n) = PM (n) ; PM (n ; e) for all n 2 Z. 0 Therefore HM=xM (t) = (1 ; te)HM0 (t). The long exact sequence of graded local cohomology derived from the exact sequence above easily yields that 00 0 00 likewise HM=xM (t) = (1 ; te )HM00 (t). By induction, HM=xM (t) = HM=xM (t), 0 00 so HM (t) = HM (t) as well. (c) follows immediately from (b) and Exercise 4.4.10. The previous theorem generalizes Hilbert's theorem 4.1.3, and consequently PM is termed the Hilbert quasi-polynomial of M . Suppose that M = R in 4.4.3 and that R is Cohen{Macaulay. Then deg HR (t) equals maxfn : H d (R )n 6= 0g, and thus is the a-invariant of R introduced in Section 3.6 see the remark following 3.6.19. This fact motivates the following extension of the notion of a-invariant: m
176
4. Hilbert functions and multiplicities
De nition 4.4.4. Let R be a positively graded k-algebra where k is a eld. Then the degree of the Hilbert function of R is denoted by a(R ) and called the a-invariant of R . Observe that a(R ) < 0 if and only if H (R n) = PR (n) for all n 0. That this condition has structural implications, is exhibited by a theorem of Flenner 107] and Watanabe 389]: if k is an algebraically closed eld of characteristic 0, and R is a normal Cohen{Macaulay positively graded k-algebra with negative a-invariant, then R has rational singularities, provided R has rational singularities for all prime ideals di erent from the maximal ideal of R . In Chapter 10 we will again encounter the condition a(R ) < 0. p
p
The Hilbert function of the canonical module. Stanley's theorem 4.4.6 analyzes how the Gorenstein property of a positively graded k-algebra is
reected by its Hilbert series. It will be deduced from the next result which asserts that the canonical module of a Cohen{Macaulay positively graded k-algebra is determined by its Hilbert series, provided R is a domain. Occasionally one can use this fact to identify the canonical module see for example 6.4.9. The automorphism ' : Zt t;1] ! Zt t;1], '(t) = t;1 , can be extended to all rational functions F (t), and we set F (t;1) = '(F (t)). Theorem 4.4.5. Let k be a eld, R a d-dimensional Cohen{Macaulay
positively graded k-algebra, M a Cohen{Macaulay graded R-module of dimension n, and M 0 = ExtdR;n(M !R ). Then (a) HM 0 (t) = ( 1)nHM (t;1), (b) if R is a domain, dim M = d, and HM (t) = tq H!R (t) for some q, then M (q) = !R .
;
Proof.
(a) We set
VM (t) =
X i2Z
dimk (H n (M );i )ti: m
By the graded local duality theorem 3.6.19 one has VM (t) = HM 0 (t). Furthermore HM (t) = VM (t;1) if dim M = 0. Let a 2 R be an M -regular homogeneous element of degree g. Then the exact sequence
0 ;! M (;g) ;! M ;! M=aM ;! 0 induces an exact sequence 0 ;! H n;1(M=aM ) ;! H n (M (;g)) ;! H n (M ) ;! 0: Since H n (M (;g)) = H n (M )(;g), one obtains VM=aM (t) = (t;g ; 1)VM (t). a
m
m
m
m
m
4.4. Hilbert functions over graded rings
177
By 1.5.11 there exists a maximal M -sequence x of homogeneous elements. Set bi = deg xi . An iterated application of the previous argument then yields H (t) V xM (t;1) n ;1 n 0 ;1 HM (t) = Qn M=xM bi = QM= n bi = (;1) VM (t ) = (;1) HM (t ): i=1(1 ; t ) i=1(1 ; t ) (b) We may assume q = 0. Then HM 0 (t) = H!R0 (t) = HR (t). It follows that there exists an element x 2 M 0 of degree 0, x 6= 0. Let ' : R ! M 0 be the homogeneous R -module homomorphism mapping 1 to x. Since R is a domain and M 0 a Cohen{Macaulay R -module of maximal dimension, the homomorphism ' is injective. But since R and M 0 have the same Hilbert series, ' must actually be an isomorphism, and it follows that M = M 00 = R0 = !R . Corollary 4.4.6 (Stanley). With the notation and hypothesis of 4.4.5 suppose P Q s d i that R has the Hilbert series HR (t) = i=0 hi t = j =1(1 ; taj ). (a) Then H!R (t) = (;1)d HR (t;1), equivalently, P P t aj ;s si=0 hs;i ti H!R (t) = Qd : aj j =1(1 ; t ) (b) If R is Gorenstein, then HR (t) = (;1)d ta(R)HR (t;1): (c) Suppose R is a domain, and HR (t) = (;1)d tq HR (t;1) for some integer q. Then R is Gorenstein.
(a) follows immediately from 4.4.5, and according to 3.6.11 we have !R = R (a(R )). This implies (b). With R = M , (c) results from 4.4.5(b). Remarks 4.4.7. (a) Assume theQpositively graded k-algebra R is Gorenstein, and write HR (t) = QR (t)= di=1(1 ; tai ). Then the functional equation 4.4.6(b) for HR (t) is equivalent to the equation QR (t) = tdeg QR QR (t;1), that is, to the symmetry of the polynomial QR (t). (b) Consider the homogeneous k-algebra R = kX Y ]=(X 3 XY Y 2 ). Then HR (t) = 1 + 2t + t2, but R is not Gorenstein. Applying 4.2.16, we derive from R a reduced non-Gorenstein Cohen{Macaulay ring S satisfying HS (t) = (;1)d tq HS (t;1). Thus for 4.4.6(c) it is essential to require that R be a domain. On the other hand, suppose the Hilbert series of the positively graded k-algebra R satis es 4.4.6(b), but R is not necessarily a domain. Instead suppose there exist a positively graded algebra S which is a Cohen{ Macaulay domain, and a homogeneous S -sequence x such that S=xS = R. Since QS (t) is symmetric if and only if QR (t) is, we conclude that R is Gorenstein. Proof.
178
4. Hilbert functions and multiplicities
In particular it follows that the above Artinian algebra cannot be the residue class ring of a homogeneous domain S by a homogeneous S -sequence. Stanley observed that the following result on numerical semigroup rings, due to Herzog and Kunz 160], can be derived easily from the previous corollary. A numerical semigroup is a subsemigroup S of the additive semigroup N such that 0 2 S and N n S is nite. The last condition is equivalent to the requirement that the greatest common divisor of all the elements of S is 1. If S is a numerical semigroup, then there exist integers 0 < a1 < < an such that S is the set of linear combinations z1 a2 + z2 a2 + + znan with zi 2 N: Any such set of integers is called a set of generators of S , and we write S = ha1 . . . an i. It is clear that a minimal set of generators of S is uniquely determined. The conductor c = c(S ) of S is de ned by c = maxfa 2 N : a ; 1 2= S g. For example, S = h4 7i has the conductor c(S ) = 18. If k is a eld, kS ] denotes the k-subalgebra of the polynomial ring kX ] generated by all monomials X a, a 2 S . Note that kS ] = kX a . . . X an ] if S = ha1 . . . ani. Thus, if we set deg X = 1, then kS ] is a positively graded k-algebra with k-basis X a, a 2 S . Moreover, kS ] is Cohen{Macaulay since it is a one dimensional domain. Theorem 4.4.8. Let S be a numerical semigroup with conductor c. The 1
following conditions are equivalent: (a) kS ] is Gorenstein (b) the semigroup S is symmetric, that is, for all i with 0 has i S if and only if c i 1 = S. Proof. Write R = kS ]. Then
2
;; 2
HR (t) =
Xj
; ; X ti
t = 1=(1 t)
j 2S
i c ; 1 one
i2NnS
;HR (t;1) = t=(1 ; t) + X t;i:
and so Suppose HR (t) = ;
tr HR
(t;1)
1=(1 ; t) ;
X
i2NnS
then necessarily r = c ; 1, and
i2NnS
;
ti = tc =(1 t) +
X
i2NnS
tc;1;i:
Hence HR (t) = ;tc;1 HR (t;1) if and only if S is symmetric, and the assertion follows from 4.4.6.
179
4.4. Hilbert functions over graded rings
A homogeneous Cohen{Macaulay k-algebra R is called a level ring if all elements in a minimal set of generators of !R have the same degree. When R is Artinian, then (!R ) = dimk Soc R , and therefore R is a level ring if and only if the homogeneous socle of R equals Rs where s = maxfi : Ri 6= 0g. Recall that a Cohen{Macaulay ring R is generically Gorenstein if R is Gorenstein for all minimal prime ideals of R . Theorem 4.4.9 (Stanley). Let R be a homogeneous Cohen{Macaulay kp
p
algebra. Suppose that P R is a domain, or generically Gorenstein and a level ring. Let HR (t) = si=0 hi ti=(1 t)d then
Xj i=0
Proof.
;
X hs;i j
hi
for all j = 0 . . . s:
i=0
Note that the least degree of a homogeneous non-zero element of
;
!R is b = a(R ). Our assumptions guarantee the existence of a homogeneous element x !R of degree b such that Rx = R ( b). This is clear if R is a domain. Next assume that R is generically Gorenstein and a level ring. Then the natural homomorphism ' : !R (!R ) (which is homogeneous) is a monomorphism see 1.4.1. Let G (!R ) be a homogeneous epimorphism, where G is free. Then the dual homomorphism (!R ) F = G is a monomorphism which, composed with ', yields a homogeneous monomorphism : !R F . We identify !R with its F . Suppose that the k-vector space (!R )b is contained in S image Fin. Then, since (!R )b F is a subspace of (!R )b , and as we may 2Ass R assume that k is in nite (the reader should check this), it follows that (!R )b F for some Ass R . However the elements of (!R )b generate the canonical module, and so !R F . This is impossible, because it would imply that SF contains a free R -module of rank 1. Now we choose x (!R )b 2Ass R F then Rx = R ( b). Thus, in any case, there exists an exact sequence of graded R -modules
; ! !
2
!
\
p
p
p
p
2
n
2
!
p
p
p
p
; p
p
p
0 ;! R ;! !R (b) ;! N ;! 0 where '(1) = x isPa non-zero homogeneous element of degree b in !R . Let HR (t) = si=0 hi ti =(1 ; t)d , d = dim R , be the Hilbert series of R . By 4.4.6, the exact sequence implies that '
HN (t) = (1
;
t);d
Xs i=0
(hs;i ; hi)ti :
The module N has rank 0 since rank !R = 1, and so dim N < d . On the other hand, the exact sequence shows that depth N d ; 1, and thus we conclude that N is a graded Cohen{Macaulay module of dimension
180 d
4. Hilbert functions and multiplicities
; 1. Therefore HN (t) = Pri=0 aiti=(1 ; t)d;1 with ai 0 for i = 1 . . . r,
see 4.1.10. It follows that
Xs
Xr
i=0
i=0
(hs;i ; hi)ti = (1 ; t)
aiti :
Thus we obtain the following set of equations: hs
; h0 = a0 hs;1 ; h1 = a1 ; a0 . . . h0 ; hs = as ; as;1:
Here we have set ai = 0 for i > r. Adding up the rst j + 1 equations gives
Xj i=0
hs;i
; X hi = aj 0 j
i=0
as asserted. Consider the sequence (1 2 1 1). Proposition 4.2.15 implies that this is the h-sequence of a Cohen{Macaulay reduced homogeneous k-algebra R . But 4.4.9 implies that such an R is not a domain. Exercises
Let F (t) = Q(t)=P di=1(1 ; tai ) = i=a fiti with Q(t) 2 Z t t 1 ] and positive integers a1 . . . ad . Let n=a fn tn be the Laurent expansion of F at 0. Show (a) there exists a unique quasi-polynomial P with P (n) = fn for n 0, (b) maxfn : fn 6= P (n)g = deg F . Hint: For (b) one argues similarly as in the proof of 4.1.12. 4.4.11. Let R be a Noetherian positively graded k-algebra over a eld k, generated by homogeneous elements x1 . . . xm of degrees e1 . . . em . Let e be the least common multiple of e1 . . . em and de ne S to be the k-subalgebra generated by the degree e homogeneous elements of R . A nite graded R -module obviously L decomposes into the direct sum of its S -submodules Mi = j Z Mje+i , i = 0 . . . e ; 1. (a) Show that the Mi are nite S -modules. (b) By considering S as a homogeneous k-algebra in the appropriate way, deduce that the Hilbert function H (M n) is a quasi-polynomial of period e for n 0. 4.4.12. Let R be a Noetherian positively graded k-algebra over a eld k and M 6= 0 a nite R -module. Furthermore let S be a graded Noether normalization of R= Ann M generated by elements of degrees a1 . . . ad , d = dim M . (a) Derive 4.4.1 from Hilbert's syzygy theorem 2.2.14 by computing the polynoQ mial Q(t) 2 Z t t 1 ] with HM (t) = Q(t)= di=1 (1 ; tai ) moreover, show Q(1) = rankS M > 0. (b) Prove that the coe cients of Q(t) are non-negative if M is a Cohen{Macaulay module. Q
4.4.10.
P
1
;
1
2
;
4.4. Hilbert functions over graded rings
181
(a) Let R be a Noetherian positively graded k-algebra of dimension 1, where k is an algebraically closed eld. If H (R n) > 1 for some n, show R is not a domain. (b) Find an example of a 1-dimensional homogeneous R-algebra R which is a domain, and for which H (R n) = 2 for all n > 0. 4.4.14. Let k be a eld, and P (t) a formal power series with integer coe cients. Demonstrate the following conditions are equivalent: (a) there exists a d -dimensional homogeneous k-algebra which is a complete intersection, and which has the Hilbert series P (t) (b) there exist an Q integer n, n d , and integers ai > 0, i = 1 . . . n, such that P (t) = (1 ; t) d ni=1 (1 + t + t2 + + tai ). 4.4.15. Let k be a eld. In this exercise we want to specify the associated prime ideals of an ideal I R = k X1 . . . Xn ] which is generated by monomials in the variables X1 . . . Xn . We order the monomials in the reverse degree-lexicographical order, and denote by L(a) the leading monomial of a see the proof of 4.2.3 and the discussion following 4.2.4.P (a) Let a 2 R , and write a = i ivi with i 2 k and vi monomials for all i. Then L(a) 2 I if a 2 I . Conclude from this that a 2 I if and only if vi 2 I for all i with i 6= 0. (b) Let a 2 R be a monomial. Show the ideal J = fb 2 R : ba 2 I g is generated by monomials. (c) Prove that an ideal generated by monomials is a prime ideal if and only if it is generated by a subset of fX1 . . . Xn g. (d) Prove that the associated prime ideals of R=I are all generated by subsets of fX1 . . . Xn g (e) Show an ideal I generated by monomials is a primary ideal if and only if it satis es the following condition: for every variable Xi which divides a monomial m 2 I such that m=Xi 2= I , some power of Xi belongs to I . 4.4.16. (a) Let k be a eld, and I1 I2 I3 k X1 . . . Xn ] ideals generated by monomials. Show I1 \ (I2 + I3 ) = (I1 \ I2 ) + (I1 \ I3 ). (b) Let v1 . . . vm k X1 . . . Xn ] monomials in X1 . . . Xn . Suppose v1 = ab is the product of monomials a and b with greatest common divisor 1, then show (ab v2 . . . vm) = (a v2 . . . vm ) \ (b v2 . . . vm). (c) Describe an algorithm to determine the primary components of an ideal generated by monomials. 4.4.17. Let k be a eld, and I k X1 . . . Xn ] an ideal generated by squarefree monomials. Demonstrate that k X1 . . . Xn ]=I is reduced. 4.4.18. Let k be a eld and I R = k X1 . . . Xn ] the ideal generated by the monomials Xi Xj , 1 i < j n. Determine Ass R=I . 4.4.19. Let k be a eld, I k X1 . . . Xn ] an ideal generated by monomials, and R = k X1 . . . Xn ]=I . Show (a) Show that a 0-dimensional Gorenstein ring is a complete intersection. (This is also true in dimension 1 see Bruns and Herzog 58].) (b) Give a 2-dimensional Gorenstein example that is not a complete intersection. 4.4.13.
;
182
4. Hilbert functions and multiplicities
Prove the graded version of 3.1.16: let R be a graded ring, and M and N graded R -modules if x 2 R is a homogeneous element of degree a which is R and M -regular and annihilates N , then ExtiR+1 (N M )(;a)
= ExtiR=(x) (N M=xM ). 4.4.21. Let k be a eld, and let R be a homogeneous Gorenstein k-algebra of dimension d . Prove 2e1 = (a(R ) + d )e0 . 4.4.20.
4.5 Filtered rings In this section we introduce the extended Rees ring and associated graded ring of a ltered ring. We will compute their dimensions, and show that a ltered ring inherits many good properties from its associated graded ring. The results will be used in the next section where we consider the Hilbert{Samuel function, and in Chapter 7 for the study of graded Hodge algebras. De nition 4.5.1. Let R be a ring. A ltration F on R is a descending chain R = I0 I1 I2 of ideals such that Ii Ij Ii+j for all i and j . A ltered ring is a pair (R F ) where R is ring and F is a ltration on R . The most common ltration is the one given by the powers of an ideal I , called the I-adic ltration. Let R be a ltered ring with ltration F = (Ii )i 0 . We de ne the extended Rees ring of R with respect to F by
R(F ) = M Iiti: i2Z
Here Ii = R for i 0, and R(F ) is viewed as a graded subring of R t t;1]. Moreover, we de ne the associated graded ring of R with respect to F by grF (R ) =
1 M i=0
Ii =Ii+1:
It is a graded ring with multiplication induced by the multiplication map Ii Ij ! Ii+j . L an R -module M , R(F M ) = i2Z Ii Mti (respectively grF (M ) = L1Given i=0 Ii M=Ii+1M ) is in a natural way a graded module over R(F ) (respectively grF (R )). In the case where F is the I -adic ltration we denote by R(I ) the extended Rees ring, and in accordance with Section 1.1 by grI (R ) the associated graded ring. Further we write R(I M ) for R(F M ) and grI (M ) for grF (M ). L We will also encounter the ReesLring R+(F ) = 1i=0 Iiti and the 1 graded R+(F )-modules R+(F M ) = i=0 Ii Mti . The notations in the case of I -adic ltrations are to be modi ed accordingly. The following observation, whose proof is left to the reader, is of crucial importance in the study of the extended Rees ring.
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4.5. Filtered rings
Lemma 4.5.2. Let R be a ltered ring with ltration F. Then (a) the element t;1 2 R t t;1] belongs to R(F ) and is R(F )-regular, (b) R(F )=t;1R(F ) = grF (R ), (c) R(F )t; = R t t;1]. We call F Noetherian if R(F ) is Noetherian. For example, I -adic ltrations on a Noetherian ring are Noetherian. It is clear that if R is Noetherian and R(F ) is nitely generated over R , then F is Noetherian. But the converse is true as well: Proposition 4.5.3. Let R be a ltered ring with ltration F = (Ii)i 0 . The 1
following conditions are equivalent: (a) F is Noetherian (b) R is Noetherian, and (F ) is nitely generated over R (c) R is Noetherian, and +(F ) is nitely generated over R (d) R is Noetherian, and there exist P positive integers j(1) . . . j(n) and xi Ij (i), i = 1 . . . n, such that Ik = ni=1 xiIk;j (i) for all k > 0.
R R
2
The equivalence of the statements (a), (b) and (c) follows from 1.5.5. (c) ) (d): Let R+(F ) = R a1 . . . an]. We may assume that a1 . . . an are homogeneous elements of positive degree. Then ai = xi tj (i) for some j (i) > 0 and xi 2 Ij (i), i = 1 . . . n. These xi satisfy the conditions in (d). (d) ) (a) is proved similarly. Note that if R=I1 is Noetherian and grF (R ) is nitely generated over R=I1, it does not follow in general that F is Noetherian. For example, let (R ) be a local ring and let F = (Ii)i 0 with Ii = for all i > 0. Then grF (R ) = R= , but R(F ) is not Noetherian. Thus we have T to pose an extra condition on F : the ltration F = ( I ) is separated if i i 0 i 0 Ii = 0, T and F is strongly separated if i 0 (I + Ii ) = I for all ideals I R . By Krull's intersection theorem, I -adic ltrations on local rings are strongly separated, provided I 6= R . Recall that the ltration F = (Ii )i 0 de nes a topology on R whose base is given by the sets a + Ii , a 2 R and i 0 see 289]. With this topology R is a HausdorTspace if and only if F is separated. The closure of an ideal I is given by i 0 (I + Ii ) hence F is strongly separated if and only if all ideals of R are closed subsets. Let us denote by M^ the completion of an R -module M with respect to F (see 289], 9.5). Then R^ is complete with respect to the ltration F^ = (I^i)i 0 , and I^i is the closure of Ii R^ in R^ . If the ltration is separated, then the canonical homomorphism R ! R^ is injective, and I^i \ R = Ii see 289], Theorem 3, p. 390. Further, if grF (R ) is Noetherian, then R^ is Noetherian and for all ideals I R , I R^ is the closure of I in R^ (289], Theorem 15 and Corollary 1, p. 413). Proof.
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Proposition 4.5.4. If F is Noetherian, then R is Noetherian and grF (R ) is
nitely generated over R=I1. Conversely, if F is strongly separated, R=I1 is Noetherian, and grF (R ) is nitely generated over R=I1, then F is Noetherian.
The rst part of the assertion is obvious in view of 4.5.3. For the converse we show that R is Noetherian, and that R(F ) is nitely generated over R . Then, by 4.5.3, F is Noetherian. Let I R be an ideal of R , and a 2 I R^ \ R . Since I R^ is the closure of I in R^ , there exist, for all i T 0, elements ai 2 I such that ai ; a 2 I^i \ R = IiR^ \ R = Ii. Therefore a 2 i 0 (I + Ii ) = I . Thus we have I R^ \ R = I , and this proves that R is Noetherian since R^ is Noetherian. In order to prove that R(F ) is nitely generated we may assume that grF (R ) = R=I1 x1 . . . x n ] where the x i are homogeneous of positive degree, say x i = xi + Ij (i)+1, xi 2 Ij (i) for i = 1 . . . n. Let A = R t;1 x1 tj (1) . . . xn tj (n)] then A is a graded subalgebra of R(F ), and we claim that indeed A = R(F ): let k > 0 and x 2 Ik tk then x = a0 + b0 tk with a0 2 Ak and b0 2 Ik+1, by the de nition of A. For the same reason we have b0 tk+1 = a1 + b1tk+1 with a1 2 Ak+1 and b1 2 Ik+2. It follows that x = a0 + a1 t;1 + b1tk , and hence x 2 Ak + Ik+2tk . T By induction on j one shows that x 2 Ak + Ik+j tk for all j 1. Thus x 2 j 1 (Ak + Ik+jtk ) = Ak since F is strongly separated. In the next theorem we compare the dimension of a module M with the dimension of R(F M ) and grF (M ) where F is a ltration on R . For the proof we will have to identify the minimal prime ideals of R(F ). Let 2 Spec R then 0 = R t t;1] \ R(F ) is a prime ideal of R(F ) and 0 \ R = . It is clear that 0 belongs to the set D(t;1) of graded prime ideals of R(F ) which do not contain t;1 . Proof.
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Lemma 4.5.5. Let F be a Noetherian ltration. (a) The map : Spec R ! D(t;1), 7! 0 , is an inclusion preserving p
bijection
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(b) height = height 0 for all 2 Spec R (c) induces a bijection between the minimal prime ideals of R and R(F ). Proof. (a) It is clear that is injective and inclusion preserving. Let 2 D(t;1) then R(F )t; = Rt t;1] is a graded prime ideal of R t t;1], and hence of the form R t t;1] for some 2 Spec R . It follows that = R(F )t; \ R(F ) = R t t;1] \ R(F ) = 0 . (b) Obviously we have height 0 height . Suppose height 0 = h. By 1.5.8, there exists a strictly descending chain of graded prime ideals 0= ;1 0 1 h . Since all i 2 D(t ), there exist i 2 Spec R 0 with i = i . Then = 0 1 h is a strictly descending chain of prime ideals in R thus height h. p
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(c) Let be a minimal prime ideal of R(F ). Then t;1 2= since t;1 is R(F )-regular. According to 1.5.6, is graded, and so belongs to D(t;1). The rest follows from (a) and (b). Theorem 4.5.6. Let R be a ltered ring with Noetherian ltration F = (Ii)i 0 , and M a nite R-module. Then R(F M ) is a nite R(F )-module, P
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(a) dim R(F M ) = dim M + 1, (b) dim grF (M ) = supfdim M : 2 Supp(M=I1M ) maximalg. In particular, dim grF (M ) dim M, and dim grF (M ) = dim M if I1 is contained m
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(a) It is clear that R(F M ) is a nite R(F )-module. Let J = Ann M , and set R 0 = R=J and F 0 = (IiR 0 )i 0 . Then M is an R 0 -module and R(F M ) = R(F 0 M ). Thus we may as well assume that M is a faithful R -module. But then R(F M ) is a faithful R(F )-module, too, so that dim R(F M ) = dim R(F ). Therefore it suces to prove the assertion for M = R . Let 2 Spec R(F ), and set = \ R . We choose a minimal prime ideal such that height( = ) = height . By 4.5.5, there exists 2 Spec R such that = 0 . Thus we obtain the nitely generated extensions R= R(F )= 0 R t t;1]= R t t;1] of integral domains, and it follows that the transcendence degree of the fraction eld Q(R(F )= 0 ) over Q(R= ) is one. Thus A.19 yields height = height( = 0 ) height( = ) + 1 height + 1. In particular we conclude that dim R(F ) dim R + 1. Conversely, dim R(F ) dim R(F )t; = dim R t t;1] = dim R + 1. The reader may check the last equality. (b) As for (a) we may reduce the assertion to the case in which M is faithful. Then R(F M ) is a faithful R(F )-module, and therefore grF (M ) = R(F M )=t;1R(F M ) is a faithful grF (R )-module. Thus we may assume M = R . According to Exercise 1.5.25, the dimension of grF (R ) is the supremum of all numbers dim grF (R ) where the supremum is taken over all graded maximal ideals 2 Spec grF (R ). Let be such an ideal and its preimage in R(F ). Then is a graded maximal ideal, and (hence) contains t;1. Let = \ R . As R(F )= is a graded ring and a eld, it is isomorphic with its degree zero homogeneous component R= . Thus Proof.
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Conversely, let I1 be a maximal ideal of R . Then R(F )=( 0 t;1) = R= , whence = ( 0 t;1) is a maximal ideal of R(F ) with = \ R . As above, it follows that height = height for the graded maximal ideal = =(t;1) of grF (R ). m
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The next series of results demonstrate that `good' properties of grF (R ) descend to R . Theorem 4.5.7. Let R be a ltered ring with Noetherian ltration F = (Ii)i 0 . (a) If grF (R ) is Cohen{Macaulay, then so is R for all 2 V (I1 ). (b) If grF (R ) is Gorenstein, then so is R for all 2 V (I1). Proof. (a) Let 2 V (I1). The ltration F = (Ii )i 0 on R induces the ltration F 0 = (IiR )i 0 on R , and we have R R grF (R ) = grF 0 (R ). Thus we may as well assume that (R ) is local, I1 , and = . Then R(F ) is local, and t;1 belongs to the unique graded maximal ideal of R(F ). Therefore R(F ) is Cohen{Macaulay;1by 4.5.2(a), (b) and Exercise 2.1.28. Applying 4.5.2(c) we see that R t t ] is Cohen{Macaulay. Since the extension R ! R t t;1] is faithfully at, R is Cohen{Macaulay see 2.1.23. (b) is proved in a similar manner. p
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Theorem 4.5.8. Let R be a ltered ring with separated ltration F. If grF (R ) is reduced or a domain, then so is R. The proof, whose details we leave to the reader, follows easily from 4.5.10. We close this section by showing that, under mild hypotheses, normality of the associated graded ring implies normality of the ring itself. Let R be a Noetherian domain with fraction eld K . Recall that x 2 K is completely integral over R if there exists an element a 6= 0 in R such that axn 2 R for all n 0. Note that R is normal (integrally closed in K ) if and only if R is completely integrally closed, i.e. every element x 2 K which is completely integral over R is an element of R . Indeed, suppose x 2 K , x = c=d , is integral over R . Then there exists an equation xm + a1xm;1 + + am;1x + am = 0 with ai 2 R , and it is clear that d m xn 2 R for all n 0. Conversely, if x 2 K such that axn 2 R for some a 2 R , a 6= 0, and all n 0, then R x] a;1R . Since a;1R is a nite R -module this implies that x is integral over R see 270], Theorem 9.1. We introduce a notation which is useful in the proof of the following theorem. Let R be a ltered ring with separated ltration F = (Ii )i 0 . For each non-zero g 2 R there exists a unique integer i 0 such that g 2 Ii n Ii+1. We set g? = g + Ii+1 and call it the initial form of g in grF (R ) of course, 0? = 0.
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Theorem 4.5.9. TLet R be a ltered ring with Noetherian ltration F = (Ii)i 0 satisfying i 0 (aR + Ii ) = aR for all a 2 R. If grF (R ) is a normal domain, then so is R.
The assumptions imply that F is separated. Hence by 4.5.8, R is a domain. Let K be the eld of fractions of R , and x = c=d an element in K which is completely integral over R . We want to showTthat c 2 Rd . It suces to prove that c 2 Rd + Ii for all i 0, since Rd = i 0 (Rd + Ii ) by assumption. We prove this by induction on i, the case i = 0 being trivial. Suppose c 2 Rd + Ii then c = ud + w, u 2 R , w 2 Ii. As x is completely integral over R there exists a 2 R , a 6= 0, such that a(x ; u)n 2 R for all n 0, and this implies a(w=d )n 2 R for all n 0. In other words, there exist elements wn 2 R such that awn = wnd n for all n 0. We have (gh)? = g? h? for all g h 2 R since grF (R ) is a domain see Exercise 4.5.10. Applied to the above equation we obtain a? (w? )n = wn? (d ? )n. This means that w? =d ? is completely integral over grF (R ). By assumption, grF (R ) is an integrally closed domain. Therefore, w? =d ? 2 grF (R ), or equivalently, w? = v? d ? for some v 2 R . Since w 2 Ii , the last equation yields w ; vd 2 Ii+1. But then c = (u + v)d + (w ; vd ), as desired. Proof.
Exercises Let R be a ltered ring with separated ltration. Show: (a) a?b? = (ab)? or a? b? = 0 (b) a?b? = (ab)? if grF (R ) is a domain. 4.5.11. Let R be a ltered ring with Noetherian ltration F = (Ii )i 0 . Prove (a) dim R dim R+(F ) dim R + 1, T (b) dim R+(F ) = dim R + 1 () I1 6 f 2 Ass R : dim R= = dim R g. 4.5.12. Let R be a ltered ring with ltration F = (Ii )i 0 . The s-th Veronese subring R(+s) (F ) of R+(F ) is again a Rees ring which is de ned by the ltration F (s) = (Isi )i 0 . Show the following conditions are equivalent if R is Noetherian: (a) R+(F ) is a nitely generated R -algebra (b) R(+s) (F ) is a nitely generated R -algebra for all s 1 (c) there exists an integer s 1 such that R(+s) (F ) is a nitely generated R -algebra (d) there exists an integer s 1 such that Ii+s = Ii Is for all i s (e) there exists an integer s 1 such that Iis = (Ii )s . L Hints: for the proof of (c) ) (a) consider the ideals Mj = i 0 Iis+j tis of R(+s) (F ), j = 0 . . . s ; 1, and for (a) ) (d) use that R+(F ) = R I1 t . . . Ir tr ] for some r 1 then choose s = (r ; 1)r!. (The implication (a) ) (d) is due to Rees 304].) 4.5.13. Let k be a eld, R = k X 6 X 7 X 15 ] and the ideal in R generated by X 6 X 7 and X 15 . Show R is Gorenstein, but gr (R ) is not even Cohen{Macaulay. 4.5.10.
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4.6 The Hilbert{Samuel function and reduction ideals Let (R ) be a Noetherian local ring, and M 6= 0 a nite R -module. In order to de ne the multiplicity of M one passes to the associated graded module gr (M ), and de nes e(M ) = e(gr (M )): To be more exible we may as well consider an ideal I such that nM IM for some n. Any such ideal is called an ideal of de nition of M . The associated graded ring grI (R ) is a homogeneous algebra, and grI (M ) is a graded grI (R )-module. De nition 4.6.1. The rst iterated Hilbert function m
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IM (n) = H1 (grI (M ) n) =
=
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Xn i=0
H (grI (M ) i)
`(I i M=I i+1M ) = `(M=I n+1M )
is called the Hilbert{Samuel function of M , and e(I M ) = e(grI (M )) the multiplicity of M with respect to I . As an immediate consequence of 4.1.6 we obtain Proposition 4.6.2. Let (R ) be a Noetherian local ring, M = 6 0 a nite m
R-module of dimension d, and I an ideal of de nition of M. Then (a) the Hilbert{Samuel function IM (n) is of polynomial type of degree d, (b) e(I M ) = limn!1 (d !=nd ) `(M=I n+1M ). Proof. By 4.5.6, we have dim M = dim grI (M ). Thus (a) follows from
4.1.6. (b) For large n we have IM (n) = (e(I M )=d !)nd +terms in lower powers of n. This yields the desired result. The polynomial MI (X ) 2 QX ] with MI (n) = IM (n) for n 0 is called the Hilbert{Samuel polynomial of M with respect to I . When I (X ). (Note that (X ) is I = , we simply write M (X ) instead of M M not the Hilbert polynomial of gr (M ).) Examples 4.6.3. (a) Let (R k) be a regular local ring of dimension d . Then the homogeneous k-algebra gr (R ) is isomorphic to the polynomial ; X +d ring kX1 . . . Xd ] see 2.2.5. Thus R (X ) = d and e(R ) = 1. (b) Let (R k) be a regular local ring, I R a proper ideal, and S = R=I . We denote by the maximal ideal of S . The canonical epimorphism " : R ! S induces a surjective homomorphism of graded k-algebras gr("): gr (R ) ! gr (S ). Indeed, let a 2 R , a 6= 0, and m
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in gr (R ) its initial form see the de nition above 4.5.9. It is clear that the homogeneous elements of gr (R ) are just the initial forms of elements of R . For a 2 j n j +1 we de ne gr (")(a?) = "(a) + j +1. Let I ? be the ideal generated by the elements a?, a 2 I . Then ? I = Ker(gr (")) because if a? = a + j +1 2 I ? , then "(a) 2 j +1. Hence there exists b 2 j +1 such that "(b) = "(a). It follows that c = a ; b 2 I , and c? = a? . The converse inclusion is obvious. We conclude that gr (S ) = kX1 . . . Xd ]=I ? . Thus S (X ) and e(S ) may be computed once I ? and its graded resolution are known see 4.1.13. Assume I = (a1 . . . am) then (a?1 . . . a?m) I ? with equality if m = 1. In general, however, we have (a?1 . . . a?m) 6= I ? (Exercise 4.6.12). Computing e(I M ) may be a painful and often impossible task. We will show that an arbitrary ideal of de nition of M may be replaced by an ideal J which is generated by a system of parameters of M such that e(J M ) = e(I M ), provided the residue class eld k of R is in nite. De nition 4.6.4. Let R be a Noetherian ring, I a proper ideal, and M a nite R -module. An ideal J I is called a reduction ideal of I with respect to M if JI n M = I n+1M for some (or equivalently all) n 0. The de nition of a reduction ideal almost immediately yields Lemma 4.6.5. Let (R ) be a Noetherian local ring, M a nite R-module, m
j +1
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I an ideal of de nition of M, and J a reduction ideal of I with respect to M. Then J is an ideal of de nition of M, and e(J M ) = e(I M ). Proof. For large n we have I n+1M = JI n M JM , and this shows that J is an ideal of de nition of M . Moreover, we get the inequalities `(M=I m+n+1M ) `(M=J m M ) `(M=I m M ) for all m 1. Thus 4.6.2 implies the assertion.
In the framework of Rees rings and Rees modules, reduction ideals can be characterized as follows. Proposition 4.6.6. Let R be a Noetherian ring, J I proper ideals of R,
and M a nite R-module. The following conditions are equivalent: (a) J is a reduction ideal of I with respect to M (b) +(I M ) is a nite +(J )-module. Proof. (a) (b): Suppose I n+1M = JI nM then +(I M ) is generated over +(J ) by the elements of degree n, and hence is nitely generated. (b) (a): We may choose a homogeneous set of generators x1 . . . xr of +(I M ). Let n be the maximal degree of the elements xi , and let x P I n+1M . There exist elements ai J bi , bi = n + 1 deg xi , such that x = ri=1 ai xi. Since ai xi J bi I n+1;biM JI nM , it follows that x JI n M . Thus we have I n+1M JI nM . The converse inclusion is trivial.
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In terms of Rees rings we now introduce an invariant which gives a lower bound for the number of generators of a reduction. De nition 4.6.7. Let (R ) be a Noetherian local ring, I a proper ideal of R , and M a nite R -module. The number m
;R
(I M ) = dim
R+(I M) = dim;grI (M)=
+(I M )=
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is the analytic spread of I with respect to M . We set (I ) = (I R ) and call it the analytic spread of I . Proposition 4.6.8. Under the hypothesis of 4.6.7 we have (J ) (I M )
for any reduction ideal J of I with respect to M. Suppose in addition that R= is in nite. Then there exists a reduction ideal J of I with respect to M such that (J ) = (I M ). m
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The module +(I M )= +(I M ) is nite over +(J )= +(J ) = i= J i which in turn is a factor ring of kX . . . X ], where m = J 1 m i 0 ; dimk J= J = (J ). Therefore dim +(I M )= +(I M ) m. This
L
Proof.
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proves the rst part of the proposition. Now let A = R+(I )= , where is the annihilator of the R+(I )-module R+(I M)= R+(I M). The ideal is graded and contains R+(I ). Consequently A is a homogeneous R= -algebra, and dim A = (I M ). Since R= is in nite, the Noether normalization theorem says that there exist elements y1 . . . yd 2 A of degree 1, d = (I M ), such that A is a nite B -module, where B = ky1 . . . yd ] see 1.5.17. It follows that R+(I M)= R+(I M) is a nite graded B-module. For each yi we choose zi 2 I such that zi is mapped to yi under the canonical map I ! R+(I )= . Let J ;= (z1 . . . zd ) then (J ) = (I M), and R+(I M )= R+(I M ) is a nite R+(J )= R+(J ) -module. Now the graded version 1.5.24 of Nakayama's lemma implies that R+(I M ) is a nite (R+(J ))-module, and this completes the proof see 4.6.6. Remark 4.6.9. Let (R k) be a Noetherian local ring, and I a proper ideal of R . Northcott and Rees 291] call an ideal J a minimal reduction of I if J is a reduction ideal of I , and J itself does not have any proper reductions, and they prove that minimal reductions exist { a fact which we will not use explicitly here. In the case where k is in nite one has the following result: let J be a reduction of I , and suppose that J is minimally generated by x1 . . . xn. Then J is a minimal reduction of I if and only if the elements x1 . . . xn are analytically independent in I and n = (I ). Recall that x1 . . . xn are analytically independent in I if whenever f (X1 . . . Xn ) is a homogeneous polynomial of degree m in R X1 . . . Xn] (m arbitrary) such that f (x1 . . . xn ) 2 I m , then all the coecients of f are in . m
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It is clear from this description and the proof of 4.6.8 that the ideal J constructed there is a minimal reduction of I when M = R . Corollary 4.6.10. Let (R ) be a Noetherian local ring with in nite residue m
class eld, M a nite R-module, and I an ideal of de nition of M. Then there exists a system of parameters x of M such that (x) is a reduction ideal of I with respect to M. In particular e(I M ) = e((x) M ).
;
We show that dim R+(I M )= R+(I M ) = dim M . This, in view of 4.6.5 and 4.6.8, implies the assertion. Note that `(I n M= I n M ) = (I nM ), and `(I nM=I n+1M ) (I n M ) `(M=IM ). Indeed, let y1 . . . ym , m = (I nM ), be a system of generators of I nM . We may L write yj = aj xj with aj 2 I n , xj 2 M . Thus there exists an epimorphism Rxj ! I nM L which yields an epimorphism R x j ! I nM=I n+1M where x j denotes the residue class of xj modulo IM . Since R x j M=IM , the desired inequality follows. We therefore obtain the inequalities ; H R+(I M )= R+(I M ) n H (grI (M ) n) `(M=IM) H ;R+(I M)= R+(I M) n: By 4.1.3 and 4.5.6, the Hilbert function H (gr ; I (M) n) is a polynomial of degree dim M ; 1 for large n, and so is H R+(I M )= R+(I M ) n by the above inequalities. Hence, if we again apply 4.1.3, the conclusion follows. Proof.
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Exercises 4.6.11. Let R be a Noetherian ring, I a proper ideal of R , M a nite R -module, M a submodule, and M a factor module of M . If J is a reduction ideal of I with respect to M , show it is a reduction ideal of I with respect to M and M , 0
00
0
00
too. (Hint: Use 4.6.6). 4.6.12. (a) Let (R k) be a regular local ring, and f an element in whose initial form f ? has degree a see 4.6.3. Set S = R=(f ) and prove that gr (S )
= k X1 . . . Xd ]=(f ? ), and e(S ) = a. (b) Let I = (X 2 XY + Z 3 ) k
X Y Z ]]. Show the ideal I ? of initial forms of I is not generated by the initial forms X 2 and XY of the generators of I . 4.6.13. Let (R ) be a Noetherian local ring and I a proper ideal of R . Show that the analytic spread of an ideal has the following properties: (a) (IR ) (I ) for all 2 Spec R (b) if I is -primary, then (I ) = dim R (c) height I (I ) dim R . 4.6.14. Let (R k) be a d -dimensional Cohen{Macaulay local ring. Prove: (a) If k is in nite, then there exists an R -sequence x = x1 . . . xd such that e(R ) = `(R=(x)). Hint: Proceed by induction on d choose x1 such that its initial form in gr (R ) is an element of degree 1 whose annihilator has nite length. m
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(b) e(R ) embdim R ; dim R +1 if equality holds, then R is said to have minimal multiplicity. (c) If k is in nite, then R has minimal multiplicity if and only if there exists an R -sequence x such that 2 = (x) . 4.6.15. Let (R k) be a one dimensional local ring. Prove: (a) If k is in nite, there exists an element x such that n+1 = x n for all n 0 any such element is called supercial. (b) If x is a super cial element, then x 2= 2 . (c) e(R ) = ( n ) for large n. 4.6.16. Let (R k) be a one dimensional Cohen{Macaulay local ring, and x a super cial element of R . (a) Suppose I is an ideal of height 1 in R . Show that `(I=xI ) = e(R ). (b) Prove that (I ) e(R ) for all ideals of height 1 of R . 4.6.17. Let (R k) be a Noetherian local ring. Suppose there exists an integer n such that (I ) n for all ideals I of R . Show that dim R = 1. 4.6.18. Let R = k
t]], k a eld. Let f1 (t) . . . fn (t) 2 R . We denote by k
f1 (t) . . . fn (t)]] the subring A = fF (f1 (t) . . . fn (t)): F 2 k
X1 . . . Xn ]]g of R . Suppose the integral closure of A is R prove e(A) is the minimum of the initial degrees of the fi (t). Hint: use the fact that R is a nite A-module see
270], x33. m
m
m
m
m
m
m
m
m
4.7 The multiplicity symbol In the previous section we saw that the computation of the multiplicity e(I M ) of a nite module M with respect to an ideal of de nition I can be reduced to the case when I is generated by a system of parameters of M . The advantage of this reduction will become apparent when we show that the multiplicity of a module M with respect to an ideal generated by a system of parameters x can be expressed in terms of the Koszul homology H (x M ). We approach this goal by introducing the multiplicity symbol e(x M ), due to Northcott. Let (R ) be a Noetherian local ring, and M a nite R -module. A sequence of elements x = x1 . . . xn in is a multiplicity system of M if `(M=(x)M ) is nite, equivalently, if (x) is an ideal of de nition of M . Lemma 4.7.1. Let (R ) be a Noetherian local ring, x a sequence of elements in R, and 0 ! M 0 ! M ! M 00 ! 0 an exact sequence of nite .
m
m
m
R-modules. The sequence x is a multiplicity system of M if and only if it is a multiplicity system of M 0 and M 00 . Proof. The exactness of M 0 =(x)M 0 M=(x)M M 00 =(x)M 00 0 im-
!
!
!
plies that `(M 00 =(x)M 00) `(M=(x)M ) `(M 00 =(x)M 00) + `(M 0 =(x)M 0 ):
193
4.7. The multiplicity symbol
It therefore remains to show that `(M 0 =(x)M 0 ) < 1 if `(M=(x)M ) is. According to the Artin{Rees lemma (270], Theorem 8.5) there exists an integer m such that (x)mM \ M 0 (x)M 0 , and this implies that `(M 0 =(x)M 0 ) `(M 0 =(x)mM \ M 0 ) `(M=(x)m M ). Corollary 4.7.2. Let (R ) be a Noetherian local ring, M a nite R-module, and x = x1 . . . xn a multiplicity system of M. Then x0 = x2 . . . xn is a multiplicity system of M=x1 M and (0 : x1 )M . Proof. Note that x(M=x1 M ) = x0 (M=x1M ), and x(0 : x1 )M = x0(0 : x1 )M . m
This corollary allows an inductive de nition of the multiplicity symbol. De nition 4.7.3. Let (R ) be a Noetherian local ring, M a nite R module, and x = x1 . . . xn a multiplicity system of M . If n = 0, then `(M ) < 1, and we set e(x M ) = `(M ) if n > 0, we set e(x M ) = e(x0 M=x1 M ) ; e(x0 (0 : x1 )M ), x0 = x2 . . . xn . We call e(x M ) the multiplicity symbol. At rst glance it seems as if the multiplicity symbol depends on the order of the elements of x. That this is not the case will follow from the next theorem. Note that the homology H (x M ) of the Koszul complex of a multiplicity system x of M has nite length as follows from 1.6.5. Hence we may consider the Euler characteristic m
.
(x M ) =
X i
(;1)i `(Hi (x M ))
of the Koszul homology. Theorem 4.7.4 (Auslander{Buchsbaum). Let (R ) be a Noetherian local ring, M a nite R-module, and x a multiplicity system of M. Then m
e(x M ) = (x M ):
The proof of the theorem is based on Lemma 4.7.5. Let (R ) be a Noetherian local ring, and x = x1 . . . xn a m
sequence of elements in . Whenever the Euler characteristic is de ned it has the following properties: (a) (x ) is additive on short exact sequences, that is, for any short exact sequence 0 M 0 M M 00 0 for which x is a multiplicity system of M, one has (x M ) = (x M 0 ) + (x M 00 ) (b) if x1M = 0, then (x M ) = 0 (c) if x1 is M-regular, then (x M ) = (x2 . . . xn M=x1M ). m
! ! ! !
194
4. Hilbert functions and multiplicities
(a) By the additivity of length, the alternating sum of the lengths of the homology modules in the long exact sequence ;! Hi (x M0) ;! Hi(x M) ;! Hi(x M00) ;! is zero. This yields the desired result. (b) Let x0 = x2 . . . xn. If x1M = 0, then Hi (x M ) = Hi (0 x0 M ) = Hi (x0 M ) Hi;1(x0 M ) for all i see 1.6.21. Thus Proof.
(x M ) =
X i
;
(;1)i `(Hi (x0 M )) + `(Hi;1(x0 M ) = 0:
(c) If x1 is an M -regular element, then Hi (x M ) = Hi (x0 M=x1 M ) by 1.6.13. This implies the assertion. Proof of 4.7.4. Let x = x1 . . . xn and x0 = x2 . . . xn. We show that (7) (x M ) = (x0 M=x1 M ) ; (x0 (0 : x1)M ):
The ascending chain 0 (0 : x1 )M (0 : x21)M of submodules of M stabilizes since M is Noetherian. Let a be an integer such that (0 : xa1 )M = (0 : xa1+1)M . We leave it to the reader to verify that x1 is regular on N = M=(0 : xa1)M , and that x0 is a multiplicity system of (0 : xa1 )M . Consider the following commutative diagram with exact rows and columns: 0 0
?? y
?? y
(0 : x1 )M
(0 : x1)M
;;;;! (0 : ?xa1)M ;;;;!
M
?? y
0
x? y 1
0
;;;;! (0 : ?xa1)M ;;;;!
0
;;;;!
?y
C
?? y 0
;;;;!
?? y
? x? y 1
??0 y
;;;;! N? ;;;;! 0 x? 1
y
M
;;;;! N? ;;;;! 0 ?
M=x1M
;;;;! N=x?1N ;;;;! 0
?? y
?? y 0
y
?y 0
195
4.7. The multiplicity symbol
From 4.7.5(a) it follows that (x0 N=x1 N ) = (x0 M=x1M ) ; (x0 C ), and (x0 C ) = (x0 (0 : x1)M ), and thus (8) (x0 N=x1 N ) = (x0 M=x1 M ) ; (x0 (0 : x1)M ): Now we apply 4.7.5(a) and (c) to see that (9) (x0 N=x1 N ) = (x N ) = (x M ) ; (x (0 : xa1 )M ): Finally, by induction on i, it follows from 4.7.5(a) and (b), and the exact sequences 0 ;! (0 : xi1;1)M ;! (0 : xi1)M ;! (0 : xi1 )M =(0 : xi1;1 )M ;! 0 that (x (0 : xi1 )M ) = 0 for all i. This, together with (8) and (9), completes the proof. If the sequence x generates the ideal (x) minimally, then the Koszul homology H (x M ) depends only on the ideal (x). By 4.7.4, the same holds for the multiplicity symbol. Much more is true: Theorem 4.7.6 (Serre). Let (R ) be a Noetherian local ring, M a nite Rmodule, x = x1 . . . xn a multiplicity system of M, and I the ideal generated .
m
by x. Then
n (x M ) = e(I M ) if x is a system of parameters of M, 0 otherwise.
Taking into account 4.7.4 we see that for any system of parameters x of M the numbers e(x M ), e((x) M ) and (x M ) are all the same. Proof of 4.7.6. Let K = K (x M ) be the Koszul complex, and for each integer m let K (m) be the subcomplex 0 ;! I m Kn ;! I m+1Kn;1 ;! ;! I m+nK0 ;! 0 of K . We rst claim that K (m) is exact for all m 0: for a xed integer i its i-cycles are Zi (K (m)) = Zi (K ) \ I m+n;iKi . By the Artin{Rees lemma (270], Theorem 8.5) we have ; Zi (K ) \ I m+n;iKi = I Zi (K ) \ I m+n;i;1Ki for all m 0. We may pick m0 large enough for this equality to hold simultaneously for all i and all m m0. P Now let m m0, and z 2 Zi (K (m) ) then z = ni=1 xi zi with zi 2 Zi(K ) \ I m+n;i;1Ki . Let e1 . . . en be a basis of K1 (x R ) with dx (ei ) = xi for iP= 1 . . . n, where dx denotes the di erential of K (x R ). Then w = ni=1 ei :zi 2 I m+n;i;1Ki+1 and dxM (w) = z . Thus K (m) is indeed exact. It follows from this, the exact sequence of complexes 0 ;! K (m) ;! K ;! K =K (m) ;! 0 .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
196
4. Hilbert functions and multiplicities
and the exactness of K , that H (K ) = H (K =K ) hence (x M ) = P n i `(H (K =K (m) )). However, since K =K (m) is of nite length for ( ; 1) i i i i=0 ; all i { its length is actually n `(M=I m+n;iM ) { we have (m)
.
.
.
.
(m)
.
.
.
i
Xn i=0
.
(;1)i `(Hi (K =K (m) )) = .
and thus for m 0,
Xn
Xn
.
i=0
(;1)i `(Ki =Ki(m))
n
(;1) i IM (m + n ; i ; 1) = n IM (m ; 1) i=0e(I M) if dim M = n, = 0 if dim M < n see 4.6.2, and use that the application of decreases the degree of a polynomial function by 1. Let (R ) be a Noetherian local ring, and I an ideal of de nition of R . We x an integer q, and denote by Kq (R ) the full subcategory of the category M(R ) of nite R -modules whose dimension is at most q. We de ne e(I M) if dim M = q, eq (I M ) = 0 if dim M < q. (x M ) =
i
m
Corollary 4.7.7. The (modi ed) multiplicity eq (I M ) is an additive function on the category Kq (R ), that is, eq (I M ) = eq (I M 0 ) + eq (I M 00 ) for all exact sequences 0 ;! M 0 ;! M ;! M 00 ;! 0 in Kq (R ). Proof. Without loss of generality we may assume that R= is in nite. For otherwise we may extend the residue class eld of R see the proof of 4.1.10. We may further assume that the module M in the above exact sequence has dimension q. By 4.6.10 there exists a system of parameters x = x1 . . . xq of M such that (x) is a reduction ideal of I with respect to M . Now 4.6.10 and 4.7.6 imply that eq (I M ) = (x M ). According to Exercise 4.6.11 the ideal (x) is a reduction ideal of I with respect to M 0 and M 00 as well. Hence we also have eq (I M 0 ) = (x M 0 ) and eq (I M 00 ) = (x M 00). Thus the result follows from 4.7.5. Corollary 4.7.8. Let (R ) be a Noetherian local ring, I an ideal of de nim
m
tion of R, and M a nite R-module of dimension eq (I M ) =
X
q. Then
`(M )eq (I R= ) p
p
p
where the sum is taken over all prime ideals
p
with dim R= = q. p
197
4.7. The multiplicity symbol
The module M has a ltration 0 = M0 M1 Mr;1 Mr = M such that Mi =Mi;1 = R= i for i = 1 . . . r. Of course, dim R= i Pq for all i. Thus by the previous corollary we have eq (I M ) = ri=1 eq (I R= i ). Only those summands contribute to the sum for which dim R= i = q. Fix a prime ideal with dim R= = q. Then the number of integers i for which = i equals the length of M , as can be easily seen by localization at . This proves the formula
Proof.
p
p
p
p
p
p
p
p
asserted. As an important special case of the previous result we have Corollary 4.7.9. Let (R ) be a Noetherian local ring, M a nite module p
p
m
of positive rank, and I an -primary ideal of R. Then e(I M ) = e(I R ) rank M: In particular, e(M ) = e(R ) rank M. Proof. Let r = rank M . By virtue of 1.4.3 we have M = R r for all prime ideals of R with dim R= = d . In particular M has maximal dimension, and so e(I M ) = ed (I M ), d = dim R . Therefore 4.7.8 yields m
p
p
e(I M ) =
p
p
X
`(M )e(I R= ) =
X
r `(R )e(I R= ) = e(I R ) rank M: p
p
p
p
p
p
Here the sums are taken over the prime ideals with dim R= = d . p
p
Partial Euler characteristics. One remarkable consequence of 4.7.4 is the following: let (R ) be a Noetherian local ring,PM a nite R -module, and x a multiplicity system of M . Then (x M ) = i ( 1)i `(Hi (x M )) 0. One de nes for all j 0 the partial Euler characteristics
;
m
j (x M ) =
X i j
(;1)i;j `(Hi (x M ))
of M with respect to x. Surprisingly, all the partial Euler characteristics are non-negative, as shown by Serre (334], Appendice II). We only prove the result for 1 (see however Remark 4.7.12). Theorem 4.7.10 (Serre). Let (R ) be a Noetherian local ring, M a nite
R-module, and x a multiplicity system of M. (a) 1(x M ) 0, or equivalently, `(M=xM ) (x M ). (b) Assume in addition that x is a system of parameters of M. Then the following conditions are equivalent: (i) 1(x M ) = 0 (ii) H1 (x M ) = 0 (iii) Hi (x M ) = 0 for i 1 (iv) x is an M-sequence (v) M is Cohen{Macaulay. m
198
4. Hilbert functions and multiplicities
Proof. Let x = x1 . . . xn . We prove (a) by induction on n: if n = 1, then 1(x1 M ) = `(H1 (x1 M )), and the assertion is trivial. Now let n > 1, and set x0 = x2 . . . xn . Notice that (x M ) = `(M=xM ) 1(x M ), whence
;
(10) 1 (x M ) = 1(x0 M=x1M ) + (x0 (0 : x1 )M ): in view of equation (7) above. By induction 1 (x0 M=x1 M ) 0, and since (x0 (0 : x1 )M ) 0, the assertion follows. (b) The equivalence of the statements (ii){(v) was shown in 1.6.19 and 2.1.2, and (iii) ) (i) is obvious. We now prove the implication (i) ) (v). Suppose that 1(x M ) = 0 then (10) implies 1 (x0 M=x1M ) = 0 and (x0 (0 : x1 )M ) = 0: By induction we may assume that M=x1M is a Cohen{Macaulay module of dimension n ; 1. It remains to show that (0 : x1)M = 0. Set M1 = M=(0 : x1 )M then the snake lemma applied to the commutative diagram 0 ;;;;! (0 : x1 )M ;;;;! M ;;;;! M1 ;;;;! 0 x1
?? y
0 ;;;;! (0 : x1 )M yields the exact sequence
x1
?? y
x1
?? y
;;;;! M ;;;;! M1 ;;;;! 0
0 ;! (0 : x1)M ;! (0 : x1 )M ;! (0 : x1 )M ;! (0 : x1)M ;! M=x1 M ;! M1=x1 M1 ;! 0: It is clear that ' is an isomorphism. We claim that = 0. Indeed, it follows from 4.7.4 and 4.7.6 that dim(0 : x1)M n ; 2 since (x0 (0 : x1)M ) = 0. On the other hand, dim R= = n ; 1 for all 2 Ass(M=x1M) since M=x1M is Cohen{Macaulay see 2.1.2. Therefore Hom((0 : x1)M M=x1M ) = 0 by 1.2.3. We obtain the isomorphisms M=x1M = M1=x1M1 and (0 : x1)M = (0 : x1 )M : It follows from (7) that 1(x M ) = `(M=xM ) ; (x0 M=x1 M ) + (x0 (0 : x1)M ) and hence the analogous equation for M1 and the isomorphisms give us 1(x M1) = 1 (x M ) = 0. Repeating these arguments we obtain a sequence of modules Mn , de ned recursively by Mn = Mn;1=(0 : x1 )Mn; with Mn=x1Mn = (0 : x1 )Mn; : = Mn;1=x1Mn;1 and (0 : x1)Mn '
1
p
p
1
1
1
199
4.7. The multiplicity symbol
Consider the composition M ! M1 ! ! Mn;1 ! Mn of the canonical epimorphisms. A simple inductive argument shows its kernel is (0 : xn1)M . Since M is Noetherian there exists an integer m such that (0 : xm1 )M = (0 : xm1 +1)M , and so the canonical epimorphism Mm ! Mm+1 must be an isomorphism therefore (0 : x1 )Mm = 0. But then, as required, (0 : x1 )M = (0 : x1 )M = = (0 : x1 )Mm = 0: Combining 4.7.9 and 4.7.10 we obtain the following Cohen{Macaulay criterion for modules: Corollary 4.7.11. Let (R ) be a Noetherian local ring, M a nite R1
m
module of positive rank, and I an ideal generated by a system of parameters of R. (a) `(M=IM ) e(I R ) rank M. (b) M is Cohen{Macaulay if and only if `(M=IM ) = e(I R ) rank M. (c) Suppose R is Cohen{Macaulay then M is Cohen{Macaulay if and only if `(M=IM ) = `(R=I ) rank M.
Remark 4.7.12. The positivity of the partial Euler characteristics can be easily proved in an important special case. Let (R k) be a Noetherian local ring containing a eld, M a nite R -module, and x a multiplicity system of M . Then j (x M ) 0 for all j 0, and if j (x M ) = 0 for some j , then Hi (x M ) = 0 for all i j . For the proof we may assume that R is complete since homology commutes with completion, so that H (x M^ ) = H (x M )b = H (x M ). The last isomorphism is valid since H (x M ) has nite length. Now, as we assume that R is complete and contains a eld, the ring R even contains its residue class eld see A.20. Let A = kX1 . . . Xn ]], and de ne a ring homomorphism ' : A ! R by '(Xi) = xi for i = 1 . . . n. We may view M as an A-module via '. It is then clear that M is a nite A-module, and that H (X1 . . . Xn M ) = H (x M ). In other words, we may assume that x is an R -sequence (replace R by A). We prove the assertions by induction on j . For j = 0 and j = 1 we know the result from 4.7.6 and 4.7.10. Now we let j > 1, and consider an exact sequence 0 ;! U ;! F ;! M ;! 0 where F is a nite free R -module. Then, owing to 1.6.11 and the assumption that x is R -regular, it follows that Hi (x M ) = Hi;1 (x U ) for all i > 1. Therefore j (x M ) = j ;1(x U ), and the proof is complete by our induction hypothesis. m
.
.
.
.
.
.
Exercises 4.7.13. (a) Let (R
) be a Noetherian reduced local ring. Verify e(R ) = where the sum is taken over all prime ideals with dim R= = dim R .
P
m
p
p
e(R= ) p
p
200
4. Hilbert functions and multiplicities
(b) Let k be a eld. Compute the multiplicity of k
X1 . . . Xn ]]=(X1 X2 X2 X3 . . . Xn 1 Xn Xn X1 ): Hint: apply 4.4.17. 4.7.14. Let (R ) be a Noetherian local ring of dimension d , M a maximal Cohen{Macaulay R -module, x a system of parameters of R , and n an integer such that n (x). Show that `(M=xM ) nd e(M ). 4.7.15. (a) Let k be a eld, and assume that R = k
X1 . . . Xn ]]=I is a domain of dimension d with quotient eld L. Let y = y1 . . . yd be a system of parameters of R . The subring A = k
y1 . . . yd ]] of R is regular and R is a nite A-module see A.22. We denote by K the quotient eld of A. (a) Show that L : K ] `(R=(y)). Equality holds if and only if R is Cohen{ Macaulay. (b) Formulate and prove a similar statement for graded k-algebras. 4.7.16. In this exercise we want to use the criterion 4.7.15 in a concrete situation. (a) Let k be a eld, and k(X1 . . . Xn ) the rational function eld in n variables over k. For a vector v = (a1 . . . an ) in Zn we set X v = X1a Xnan . If v1 . . . vn are vectors in Zn , show that
k(X1 . . . Xn ) : k(X v . . . X vn )] = j det(v1 . . . vn )j: Hint: use the theory of elementary divisors. (b) Let m1 . . . mr be monomials in X1 . . . Xn , and consider the subring R = k m1 . . . mr ] of the polynomial ring k X1 . . . Xn ]. (Such a ring is called an a ne semigroup ring and will be studied more systematically in Chapter 6.) Assume that (i) Q(R ) = k(X1 . . . Xn ), (ii) there are monomials w1 . . . wn 2 R such that `(R=(w1 . . . wn )R ) < 1. Let wi = X vi for i = 1 . . . n prove that R is a Cohen{Macaulay ring if and only if j det(v1 . . . vn )j equals the number of all monomials in R not belonging to the ideal of monomials (w1 . . . wn ). (c) Apply this criterion to show that the ring k X 3 Y X 2 Y XY XY 2 XY 3 ] is Cohen{Macaulay, but that k X 3 X 2 Y XY XY 2 Y 3 ] is not. 4.7.17. Let (A ) be a regular local ring, I an ideal and R = A=I . The R -module I=I 2 is called the cotangent module of R . (a) Suppose R is Cohen{Macaulay and generically a complete intersection, that is, R is a complete intersection for all minimal prime ideals 2 Spec R . Prove that rank I=I 2 = height I . (b) Let B be a local ring, and J B a proper ideal. The pair (J B ) is called an embedded deformation of (I A) if there exists a B -sequence x such that A
= B=xB , I = JA, and x is a B=J -sequence too. Suppose dim A=I = 0 and embdim A=I = n. If (I A) has an embedded deformation (J B ) such that B=J is generically a complete intersection, show `(I=I 2 ) n `(A=I ). (c) Let k be a eld, and I k X1 . . . Xn ] an ideal generated by monomials containing a power of each indeterminate. Show the number of monomials not contained in I 2 is greater than or equal to n + 1 times the number of monomials not contained in I . Equality holds if I is generated by powers of the Xi . ;
m
m
1
1
m
m
p
p
Notes
201
Notes die Theorie der algebraischen Formen' 171] In his famous paper `Uber published a century ago, Hilbert proved that a graded module over a polynomial ring has a nite graded free resolution, and concluded from this fact that the function (which we now call the Hilbert function) is of polynomial type. The inuence of this paper on commutative algebra has been tremendous. Till today both free resolutions and Hilbert functions have fascinated mathematicians, and many problems still remain open. For many applications it is more convenient to consider the so-called Hilbert series of a graded module. This point of view is stressed in Section 4.1. Stanley calls the ( nite) coecient vector of the numerator of this rational function the h-vector of the module. Its signi cance became apparent in Stanley's work on combinatorics. An introduction to this aspect of commutative algebra is given in Stanley's monograph 363] which is well-known as the `green book'. Certainly Stanley's work initiated a new interest in Hilbert functions other important motivations come from algebraic geometry. Graded free resolutions determine the Hilbert function, but the converse is not true, except when the module has a pure resolution. This is the content of 4.1.15 which is taken from Herzog and Kuhl 158] and Huneke and Miller 217]. Section 4.2 is based on the paper 357] in which Stanley states Macaulay's theorem on Hilbert functions in the form presented in this book. We also took 4.2.15 and 4.4.6 from this article. The latter result is Stanley's beautiful theorem characterizing graded Gorenstein domains by their Hilbert function. (A generalization of 4.4.5 has recently been proved by Avramov, Buchweitz, and Sally 26]. Theorem 4.4.9 appears in an article of Stanley 362] with the hypothesis that R be a domain. Our slightly more general version was given by Hibi 170]. Macaulay's article `Some properties of enumeration in the theory of modular systems' 263] appeared in 1927, and has become a source of inspiration in commutative algebra and combinatorics see for instance Sperner 354], Whipple 394], Clements and Lindstrom 70], Elias and Iarrobino 96], Stanley 357], Hibi 167], and Green 138]. In the rst part of his paper Macaulay shows that the Hilbert function of a homogeneous ring arises as the Hilbert function of a polynomial ring modulo an ideal which is de ned by monomials. For his proof Macaulay ordered the monomials, and thereby introduced implicitly (and possibly for the rst time) what nowadays is called a Grobner basis. Buchberger 62] was the rst to describe an algorithm computing the Grobner basis of an ideal. Robbiano 308] is an `early' survey of this topic. Meanwhile e ective computation has become an important area of research in commutative algebra, and we recommend especially the
202
4. Hilbert functions and multiplicities
books of Eisenbud 91] and Vasconcelos 384] for a detailed account. The importance of the Castelnuovo{Mumford regularity has been briey indicated by Theorem 4.3.1, which is due to Eisenbud and Goto 93]. More information is provided by the survey of Bayer and Mumford 38] and by Eisenbud's book 91]. Macaulay's main result in 263] however is the inequality H (R n +1) H (R n)hni which characterizes the Hilbert functions of homogeneous k-algebras. As a note preceding it Macaulay writes: `This proof of the theorem which has been assumed earlier is given only to place it on record. It is too long and complicated to provide any but the most tedious reading.' We present Green's proof 138] of Macaulay's theorem which is less computational than the original. The proofs of Gotzmann's theorems 136] have also been drawn from Green 138]. Theorems of Gotzmann type for exterior algebras have recently been proved by Aramova, Herzog, and Hibi 12]. The lexsegment ideals that appear in the proof of Macaulay's theorem have a remarkable `extremal' property: if J is the lexsegment ideal with the same Hilbert function as a given ideal I , then each graded Betti number ij (I ) is bounded above by ij (J ). This was shown independently by Bigatti 42] and Hulett 207] in characteristic 0 and by Pardue in positive characteristic 296]. (Macaulay's theorem states this inequality for i = 0!) See Valla 377] for a related result. In his article 320], Samuel laid the foundation of modern multiplicity theory. He was the rst to apply Hilbert's theory to the associated graded ring of an -primary ideal I in a Noetherian local ring (R ). This led to the so-called Hilbert{Samuel function, and provided the de nition of the multiplicity of R with respect to I . In this context the notion of reduction ideals, invented and investigated by Northcott and Rees in 291], plays an important role. Our Proposition 4.6.8, though formulated for modules, is taken from this paper. In a special case 4.6.8 says that the multiplicity of a module with respect to an ideal equals the multiplicity of the module with respect to a suitable system of parameters. This had already been observed by Samuel in 320]. More information on reduction ideals and related questions can be found in Sally's book 319]. As a measure of the complexity of an ideal I may serve the analytic deviation (I ) ; height I . Ideals of analytic deviation zero are called equimultiple. The interested reader may consult the monograph by Herrmann, Ikeda, and Orbanz 155]. Ideals with small analytic deviation have been studied by Huckaba and Huneke 205], 206], and Vasconcelos 383]. The question of when the Rees ring or the associated graded ring of an ideal is Cohen{Macaulay has been of central interest in commutative algebra. The problem is well understood for ideals generated by dsequences. This notion, introduced by Huneke, generalizes the notion of a regular sequence considerably, but still guarantees that the Rees ring m
m
Notes
203
of an ideal I generated by a d -sequence is isomorphic to the symmetric algebra of I see Huneke 209] and Valla 376]. The reader who wants more information on d -sequences is referred to the articles by Huneke 212], and Herzog, Simis, and Vasconcelos 161], 162]. Other approaches to the Rees ring and associated graded ring of an ideal can be found in the papers by Bruns, Simis, and Trung 60], Eisenbud and Huneke 94], Goto and Shimoda 132], Huneke 211], Ikeda 223], 224], Trung and Ikeda 225], Valla 375] and Vasconcelos 381]. A comprehensive account of the recent developments in this area is given in Vasconcelos' monograph 382]. Most important is Serre's theorem 4.7.6 which relates the multiplicity to the Euler characteristic of the Koszul complex. Serre proved this result in the mid- fties. The notes 334] by Gabriel of Serre's course at the Coll!ege de France were published 1965. Auslander and Buchsbaum, in their classic paper 19], proved a version of Serre's theorem for arbitrary Noetherian rings, and gave an axiomatic description of the multiplicity. In Section 4.7 we follow this axiomatic approach, and introduce the multiplicity symbol. This terminology stems from Northcott who, in his book 289], systematically developed multiplicity theory from the formal properties of this symbol. Corollary 4.7.11 is taken from 19]. In our presentation it is a consequence of the fact that the rst truncated Euler characteristic 1 of the Koszul complex is non-negative. Serre 334] proves this not just for the rst but also for the higher truncated Euler characteristics. We only show the non-negativity of 1 (see 4.7.10), following Lichtenbaum 258]. In writing this part of the section we consulted the article 346] of Simis and Vasconcelos. The Koszul homology can be interpreted as a Tor of modules, and this leads to a far reaching generalization: the intersection multiplicity of modules introduced by Serre 334] see Remark 9.4.8.
Part II
Classes of Cohen{Macaulay rings
205
5 Stanley{Reisner rings
This chapter is an introduction to `combinatorial commutative algebra', a fascinating new branch of commutative algebra created by Hochster and Stanley in the mid-seventies. The combinatorial objects considered are simplicial complexes to which one assigns algebraic objects, the Stanley{ Reisner rings. We study how the face numbers of a simplicial complex are related to the Hilbert series of the corresponding Stanley{Reisner ring. This is the basis of all further investigations which culminate in Stanley's proof of the upper bound theorem for simplicial spheres. It turns out that most of the important algebraic notions introduced in the earlier chapters, such as `Cohen{Macaulay', `Gorenstein', `local cohomology', and `Hilbert series', are the proper concepts in solving purely combinatorial problems. Other applications of commutative algebra to combinatorics will be given in the next chapter. 5.1 Simplicial complexes The present section is devoted to introducing the Stanley{Reisner ring associated with a simplicial complex, and studying its Hilbert series. The most important invariant of a simplicial complex, its f -vector, can be easily transformed into the h-vector, an invariant encoded by the Hilbert function of the associated Stanley{Reisner ring. It is of interest to know when a Stanley{Reisner ring is Cohen{Macaulay, because then the results about Hilbert functions of Chapter 4 may be employed to get information about the f -vector. In concluding this section we show that the Stanley{ Reisner ring of a shellable simplicial complex is Cohen{Macaulay, and study systems of parameters of such a ring. De nition 5.1.1. Let V = fv1 . . . vn g be a nite set. A ( nite) simplicial complex on V is a collection of subsets of V such that F 2 whenever F G for some G 2 , and such that fvi g 2 for i = 1 . . . n. The elements of are called faces, and the dimension, dim F , of a face F is the number jF j ; 1. The dimension of the simplicial complex is dim = maxfdim F : F 2 g: Note that the empty set is a face (of dimension ;1) of any nonempty simplicial complex. Faces of dimension 0 and 1 are called vertices 207
208
5. Stanley{Reisner rings
and edges, respectively. The maximal faces under inclusion are called the facets of the simplicial complex. Given an arbitrary collection fF1 . . . Fm g of subsets of V there is a (unique) smallest simplicial complex, denoted by hF1 . . . Fm i, which contains all Fi . This simplicial complex is said to be generated by F1 . . . Fm . It consists of all subsets G V which are contained in some Fi . A simplicial complex generated by one face is called a simplex. Each simplicial complex has a geometric realization as a certain subset (composed of simplices) of a nite dimensional ane space. This explains the geometric terminology introduced above. Geometric realizations will be discussed in the next section. As an example consider the octahedron with vertex set fv1 . . . v6 g (Figure 5.1). Its facets are the sets fv1 v3 v4 g, v2
v6
...... ..... ..... .... .. ........ ..... .. ............. ... .... .... .... . . . .... .... ... .. ... .... .... .. ... .... .... .. ... ... .... . . . . . . .... ....... .. ... . . ..... . .... . . . ... ... . ... . . . . . ...... ..... ........ ... . . . . . . . . . . . . . . . . . . . . . . . . . . ... . ..... ....... .. ... .. .... ......... ...... ..... ..... ..... .... ... .. .. . .. .. . .... ... .. .. . .... .. .. . ... . ... ... ... ... ..... ... .. . ... ... ... ... .. .. .. . . . . ... ... .. ... ... . . ... . . .... ... ...... .. ... .. ... ... ... .. .. . ... . .. ... .... .. .... ... ..... . ... ... ..... . ... .... ... ... . . .. ... ............................... ... . . . . . . . . . . . . . . . . . . ... .. . . . . .. ..... . ................................................. . . .. ... . . .... .. . . ........................................ ... . . ... ....... .. ....... ... ....... ...... ....... .. .. ...... ........ ....... . .. ... ........... ....... . . ....... ......... ....... ....
v5
v3
v4
v1
Figure 5.1
fv1 v4 v5g, fv1 v5 v6g, fv1 v3 v6g, fv2 v3 v4g, fv2 v4 v5g, fv2 v5 v6g, and fv2 v3 v6g. An important class of simplicial complexes arises from nite sets with partial order , called posets for short. The order complex ( ) of a
poset is the set of chains of . Recall that a subset C of is a chain if any two elements of C are comparable. Obviously, ( ) is a simplicial complex. For example, if we order the elements of the set fv1 . . . v5 g according to Figure 5.2, then the order complex of the corresponding poset has the facets fv1 v2 v3 v5 g and fv1 v2 v4 v5 g.
Stanley{Reisner rings and f-vectors. Now let be an arbitrary simplicial complex of dimension d ; 1 0 on a vertex set V . We denote by fi the number of i-dimensional faces of . We have f0 = jV j, and f;1 = 1 since 2 . The d -tuple f () = (f0 f1 . . . fd ;1)
209
5.1. Simplicial complexes v5
v3
.. ..... .... ..... ... . . ... .... .... .... .... .... ....
v2
.... ..... .... ..... .... ..... .... .... . . . ..... ... . . . . ....
v1 .... ... .. .... ... .... .
v4
Figure 5.2
is called the f-vector of . For example, the octahedron has the f -vector (6 12 8), while the above order complex has the f -vector (5 9 7 2). The possible f -vectors of simplicial complexes have been determined by Kruskal 243] and Katona 232]. Given two integers a d > 0, let k(d ) k(d ; 1) k(j) a= + + + d d;1 j k(d ) > k(d ; 1) > > k(j ) j 1, be the unique d -th Macaulay representation of a see 4.2.6 and the de nition following 4.2.7. We set k(d ) k(d ; 1) a(d ) = + + + k(j ) : d +1 d j+1 Then (f0 f1 . . . fd ;1) 2 Zd is the f -vector of some (d ; 1)-dimensional simplicial complex if and only if 0 < fi+1 fi(i+1) 0 i d ; 2: However, if we consider more restricted classes of simplicial complexes, for instance those simplicial complexes whose geometric realization is a sphere, new constraints appear this will be the topic of the next sections. It turns out that the Stanley{Reisner rings are the appropriate tool to attack these problems. De nition 5.1.2. Let be a simplicial complex on the vertex set V = fv1 . . . vng, and k a ring. The Stanley{Reisner ring (or face ring) of the complex (with respect to k) is the homogeneous k-algebra k] = kX1 . . . Xn]=I where I is the ideal generated by all monomials Xi Xi Xis such that fvi vi . . . vis g 2= . The choice of the letter k in the de nition indicates that, with a few exceptions, we usually have in mind a eld for the coecient ring of a Stanley{Reisner ring. 1
1
2
2
210
5. Stanley{Reisner rings
Note that I is generated by squarefree monomials. On the other hand, if I (X1 . . . Xn)2 is any ideal which is generated by squarefree monomials, then kX1 . . . Xn]=I = k] for some simplicial complex . The correspondence between simplicial complexes and squarefree ideals is inclusion reversing: if and 0 are simplicial complexes on the same vertex set, then 0 () I0 I . Throughout this chapter we will assume, unless otherwise stated, that V = fv1 . . . vn g is the vertex set of the simplicial complex . Example 5.1.3. Let P = fv1 . . . vn g be a poset, and the order complex of P . Then I is generated by all monomials Xi Xj for which vi and vj are incomparable. In the above example, I = (X3X4 ). The dimension of a Stanley{Reisner ring can be easily determined. Theorem 5.1.4. Let be a simplicial complex, and k a eld. Then I =
\
P
F
F
where the intersection is taken over all facets F of , and F denotes the (prime) ideal generated by all Xi such that vi F. In particular,
62
P
dim k] = dim + 1: Proof. By Exercise 4.4.17, k] is reduced, and hence I
is the intersection of its minimal prime ideals by Exercise 4.4.15, all these ideals are generated by subsets of fX1 . . . Xng. Let = (Xi . . . Xis ) notice that I if and only if fv1 . . . vn g n fvi . . . vis g is a face of , and that is a minimal prime ideal of I if and only if fv1 . . . vn g n fvi . . . vis g is a facet. A simplicial complex is pure if all its facets are of the same dimension, namely dim , and is called a Cohen{Macaulay complex over k if k] is a Cohen{Macaulay ring. We say that is a Cohen{ Macaulay complex if is Cohen{Macaulay over some eld. According to Exercise 5.1.25, is a Cohen{Macaulay complex over every Cohen{ Macaulay ring k if and only if Z] is Cohen{Macaulay. As a consequence of 2.1.2 and the previous theorem we obtain Corollary 5.1.5. A Cohen{Macaulay complex is pure. We are going to relate the f -vector of a simplicial complex to the Hilbert series of k]. To this end we introduce a Zn-grading or ne grading on k]. More generally, let (G +) be an Abelian group. L A G-graded ring is a ring R together with a decomposition R = a2G Ra (as a Z-module) such that RaRb Ra+b for all a b 2 G. P
P
1
1
P
1
211
5.1. Simplicial complexes
Similarly one de nes a G-graded R -module, the category of G-graded R -modules, G-graded ideals etc. simply by mimicking the corresponding de nitions for graded rings and modules with G = Z see Section 1.5. If M is a G-graded R -module, then x M is homogeneous (of degree a G) if x Ma , and we set deg x = a. Example 5.1.6. The polynomial ring R = kX1 . . . Xn] has a natural Zn-grading:a for a = (a1 . . . an) Zn, ai 0 for i = 1 . . . n, we let Ra = cX : c k be the a-th homogeneous component of R , and set Ra = 0 if ai < 0 for some i. Here, X a = X1a Xnan for a = (a1 . . . an). Note that the Zn-graded ideals in R are just the ideals generated by
2
2
f
2
2
2 g
1
monomials, and the Zn -graded prime ideals are just the nitely many ideals which are generated by subsets of fX1 . . . Xng. Let I R be an ideal generated by monomials. Since I is Zn graded, the factor ring R=I inherits the natural Zn -grading given by (R=I )a = Ra =Ia for all a 2 Zn. In particular, Stanley{Reisner rings are Zn-graded in this way. Now let R be an arbitrary Zn-graded ring, and M a Zn -graded R module. Each homogeneous component Ma of M is an R0-module. Just as for Z-graded modules we de ne the Hilbert function H (M ): Zn ! Z by H (M a) = `(Ma ), provided all P homogeneous components of M have nite length, and call HM (t ) = a2Zn H (M a)t a the Hilbert series of M . Here t = (t1 . . . tn ) where the ti are indeterminates, and t a = ta1 tann for a = (a1 . . . an). For example, the Zn -graded polynomial ring R = kX1 . . . Xn] has the Hilbert series 1
HR (t ) =
X
a2Nn
ta =
Yn i=1
(1 ; ti );1:
Let us return to Stanley{Reisner rings. Given a simplicial complex , we denote by xi the residue classes of the indeterminates Xi in k] then k] = kx1 . . . xn]. We de ne the support of an element a 2 Zn to be the set supp a = fvi : ai > 0g. If xa and xb area non-zero monomials (with non-negative exponents) in x1 . . . xn, then x = xb if and only if a = b. Therefore, without ambiguity, we may set supp xa = supp a for any non-zero monomial. Note that xa 6= 0 if and only if supp a 2 , and that the non-zero monomials xa form a k-basis of k]. Therefore, X a X X a Hk] (t ) = t = t:
P
a2Nn
supp a2
F 2
a2Nn
supp a=F
P
a a If supp a=F t = 1, and if F 6= , then supp a=F t = Q F =ti=(1, ;then ti ). Thus, if we understand that the product over an empty vi 2F
212
5. Stanley{Reisner rings
index set is 1, we get Hk] (t ) =
(1)
XY
ti
1 ; ti : F 2 vi 2F
We are actually interested in the Hilbert series of k] as a homogeneous Z-graded algebra. Note that for all i 2 Z we have k]i =
M
a2Zn jaj=i
k]a
where jaj = a1 + + an for a = (a1 . . . an). (This relation explains the alternative terminology ` ne grading' for `Zn -grading'.) It follows that the Hilbert series of k] with respect to the Z-grading is obtained from (1) by replacing all ti by t. Thus we have shown Theorem 5.1.7. Let be a simplicial complex with f-vector (f0 . . . fd ;1).
Then
Hk] (t) =
X
d ;1
fiti+1 (1 t)i+1 : i=;1
;
From the Hilbert series of k] we can read o its Hilbert function: 1 if n = 0, H (k] n) = Pd ;1 f ;n;1 if n > 0. i=0 i i We note the following interesting fact: H (k] n) is a polynomial function for n > 0, and hence coincides with the polynomial for all n 0 P Hilbert ;1 f ;n;1 at n = 0 gives except possibly for n = 0. Evaluating di=0 i i () =
X
d ;1 i=0
(;1)i fi
the so-called Euler characteristic of . Thus the Hilbert function and the Hilbert polynomial of agree for all n 0 if and only if () = 1. The geometric signi cance of the Euler characteristic will become clear in 5.2.17. Two other conclusions can be drawn from 5.1.7. First, we recover that dim k] = d since the degree of the Hilbert polynomial Pk](t) is d ; 1 secondly, we see that the multiplicity of k] equals fd ;1, the number of (d ; 1)-dimensional facets of . The h-vector. Recall from 4.1.8 that a homogeneous k-algebra R of dimension d has a Hilbert series of the form HR (t) = QR (t)=(1 t)d where
;
213
5.1. Simplicial complexes
QR (t) is a polynomial with integer coecients. Let be a simplicial
complex, and write
h0 + h1t + : (1 t)d The nite sequence of integers h() = (h0 h1 . . .) is called the h-vector of . Hk] (t) =
;
A comparison with 5.1.7 yields Lemma 5.1.8. The f-vector and h-vector of a (d ; 1)-dimensional simplicial complex are related by
X i
hiti =
Xd i=0
fi;1ti (1
; t)d;i:
In particular, the h-vector has length at most d, and for j = 0 . . . d, hj =
Xj
; i fi;1 and fj;1 = Xj d ; i hi: (;1)j ;i dj ; i j ;i i=0 i=0
Comparing the coecients in the polynomial identity gives the formula for the hj in terms of the fi. In order to prove the inverse relation replace t by s=(1+ s). Then the above polynomial identity transforms into
Proof.
Xd i=0
hi si (1 + s)d ;i =
Xd i=0
fi;1si
from which one obtains the last set of equations. The octahedron has f -vector (6 12 8). Applying 5.1.8 we see that its h-vector is (1 3 3 1). We single out some special cases of the above equations: Corollary 5.1.9. With the assumptions of 5.1.8 one has h0 = 1 h1 = f0
; d
;
;
hd = ( 1)d ;1( () 1) and
Xd i=0
hi = fd ;1:
Since the f -vector and h-vector of a simplicial complex determine each other, bounds for the h-vector implicitly contain certain constraints for the f -vector. We treat an important case: Theorem 5.1.10. Let be a (d ; 1)-dimensional Cohen{Macaulay complex with n vertices and h-vector (h0 . . . hd ). Then n ; d + i ; 1 0h 0 i d: i
i
214
5. Stanley{Reisner rings
Proof. Let R
= k], where k is a eld for which k] is Cohen{Macaulay. We may assume that k is in nite. Then, since R is Cohen{Macaulay, there exists an R -sequence x of elements of degree 1 such that R = dim R=xR is of dimension 0 see 1.5.12. Now it follows from 4.1.11 that hi = H (R i) for all i. This implies already that hi 0 for all i. Notice that R is generated over k by n ; d elements of degree 1. Therefore, the Hilbert function of R is bounded by the Hilbert function of a polynomial ring in just as many variables. This yields the second inequality. To illustrate the theorem consider the simplicial complex in Figure 5.3 with facets F1 = fv1 v2 v3 g and F2 = fv1 v4 v5 g. We have f () = v3
v5
........ ........ ... . ........ ........ .. .. . . . ............ ..... . . .. ........ . . . . . .... ... . . . ........ . . . . . . ... . . . . ......... ...... . . . . .. ... . . . . . . . . . . .......... ....... . . . . . .. ... ....... . . . . . . .. ...... . . . . . . . . ..... . . . . . . . . . . . . . . ............ ....... . . . . . . . ... .... . . . . . . . . . . . . . . . . . ...................... . . . . . . . . . . . . . . . . . ... ..... . . . . . . . . .. ... . . . . . . . . . ......... ...... . . . . . . . .. ........ . . . . . . .. ... . . . . . . . . . . ............. ... . . . . . ...... ....... . . . . . .. ........ . . . . . . . . ... .... . . . . . . .............. ....... . . . .. ..... . . .. .... . . . ............. ....... . .. ... . ......... ........ . ..... .....
v1
v2
v4
Figure 5.3
(5 6 2), and so h() = (1 2 ;1). It follows that is not a Cohen{ Macaulay complex. Shellable simplicial complexes. The previous theorem will be of real use
only when we are able to exhibit interesting classes of Cohen{Macaulay complexes. Such a class is given in the following de nition. De nition 5.1.11. A pure simplicial complex is called shellable if one of the following equivalent conditions is satis ed: the facets of can be given a linear order F1 . . . Fm in such a way that (a) hFi i\hF1 . . . Fi;1i is generated by a non-empty set of maximal proper faces of hFi i for all i, 2 i m, or (b) the set fF : F 2 hF1 . . . Fi i F 62 hF1 . . . Fi;1 ig has a unique minimal element for all i, 2 i m, or (c) for all i j , 1 j < i m, there exist some v 2 Fi n Fj and some k 2 f1 2 . . . i ; 1g with Fi n Fk = fvg. A linear order of the facets satisfying the equivalent conditions (a), (b), and (c) is called a shelling of . Let us check that these conditions are indeed equivalent: (a) ) (b): We may assume that Fi = fv1 . . . vm g, and that hFi i \ hF1 . . . Fi;1i is generated by the faces fv1 . . . vj;1 vj+1 . . . vm g, 1 j r m. The unique minimal element in the set Si = fF : F 2 hF1 . . . Fii F 62 hF1 . . . Fi;1ig is fv1 . . . vr g.
215
5.1. Simplicial complexes
(b) ) (c): Let G be the unique minimal element in Si . Since G 6 Fj , there exists v 2 G n Fj . Then v 2 Fi n Fj , and it follows from the de nition of G that there exists a k, 1 k i ; 1, such that Fi n Fk = fvg. (c) ) (a): Let F 2 hFi i \ hF1 . . . Fi;1i. Then F Fj for some j < i. Let v 2 Fi n Fj as in (c). Then Fi n fvg is a maximal proper face of hFi i belonging to hFi i \ hF1 . . . Fi;1i and containing F . This proves (a). In Figure 5.4 the rst simplicial complex is shellable, the second is not. Shellable simplicial complexes arise naturally in geometry see Section
......... . ........ .. . ........ ........ ... ... . ...... ...... . . .. ... . . . . . ........... ....... . . . ........ . . . . . . . .... . . . .... . . . . ........... . . ...... . . . . . .. ... . . . . . . . . . . .......... ....... . . . . . . ... . . . . . . ......... ....... . . . . . . . .. ... . . . . . . . . . . . . . . . ......... ............ . . . . . . . . . . . . . . . .... .. . . . . . . . . . ............... . . . . . . . . ... .. . . . . . . . ....... ..... . . . . . . . . . .. . . . . . . . ...... ...... . . . . . . . .. ....... . . . . . . .. .... . . . . . . . . . . ............. ...... . . . . . .. ........ . . . . .. .... . . . . . . . ............ ....... . . . .. .. . . ......... ......... . . .. .. . . ...... ........ . . .. ........ .. .... ............. ....... ...
................................................................................. ... ... . . . . . . . . . . ....... ... . .... . . . . . . . . . ... ... .... . ... . . . . . . . . . ... . ... .. . . ... . . . . . . . . . . . . ... .. ... . . . . ...... . . . . . . . . . . . . . . ....... . . . . . ....... . . . . . ... . . . . . . .. . . . . .... . . . . .. . . . . . . .... . . . . . ... .. . . . . . .. . . . . . . .. . . . . . ... ... . . . . . ..... . . . . . .... . . . . . . .... .. ... . . . . . . . . . . . . ..... . . . . . . ...... . . . . . . . . . . . . . ..... .. . . . . . . . . . . . .. . . . . . . .. . . . . . . . ... . . . . . . . . . . . ... ... . . . . . . . . ..... . . .... . . . . . . . . . . . .... ... . . . . . . . . . .. . .... . . . . . . . . . ... .. . . . . . . . . . . . . . . . . . . . ....... ..... . . . . . . . . . . . . . . . . . . . ...... . .......................................................................................................................................................................
Figure 5.4
5.2. Other interesting classes arise from order complexes of certain posets. Here we discuss one important case. For this we need to introduce some more terminology: a ( nite) poset is said to be bounded if it has a least and a greatest element, denoted 0^ and 1^ . The poset is pure if all maximal chains have the same length, and graded if it is bounded and pure. In this case all unre nable chains between two comparable elements have the same length (Exercise 5.1.18). Let be a poset, and v 2 . The rank of v, rank v, is de ned to be the maximal length of all chains descending from v. The length of is the maximal rank of an element of . Let u v 2 we say that v covers u, written u v, if u < v, and if there is no w 2 such that u < w < v. The poset is locally upper semimodular if whenever v1 and v2 cover u, and v1 v2 < v for some v 2 , then there is t 2 , t v, which covers each of v1 and v2 see Figure 5.5.
v
v1
.. .. .. . .. ... .. . ... ... . .. ... ... .. . ... ... .. ... ... .. .. ... . . .. ... .... . . . . . . .... ... ...... . .... .. .... .. ... ...... .... . .. .... . ........ . .... .... .... .... .... .... .... . . . .... .. .... ........ .
t
u
v2
Figure 5.5
Theorem 5.1.12 (Bjorner). The order complex of a bounded, locally upper semimodular poset is shellable.
216
5. Stanley{Reisner rings
In a rst step we shall prove that for every graded poset ( ) the set of maximal chains can be given a linear order such that for any two chains m m0 2 , m : 0^ = x0 x1 xn = 1^ and m0 : 0^ = y0 y1 yn = 1^ with xi = yi for i = 0 1 . . . e, and xe+1 6= ye+1 the following conditions are satis ed: (i) If fy0 y1 . . . ye+1g is contained in a maximal chain m00 and if m0 0
Xr i=1
i = 1:
De nition 5.2.8. Let be a simplicial complex on the vertex set V . Suppose the map : V ! Rd satis es the following conditions: (a) is injective, (b) the elements of (F ) are anely independent for all F 2 , (c) relint(conv (F )) \ relint(conv (G)) = for all F G 2 , F 6= G. S Then F 2 relint(conv (F )) is called a geometric realization of .
226
5. Stanley{Reisner rings
Giving a geometric realization of its natural topology as a subspace of Rd , we note that any two geometric realizations of are homeomorphic, and we denote the underlying topological space by jj. A geometric realization always exists. Indeed, if V = fv1 . . . vn g is the vertex set of , and x1 . . . xn are anely independent elements in Rd , then : V ! Rd with (vi) = xi for i = 1 . . . n de nes a geometric realization of . Cyclic polytopes. Consider the algebraic curve M
metrically by
2
Rd , de ned para-
x() = ( 2 . . . d ) R M is called the moment curve. It is a curve of degree d which implies that a hyperplane not containing M intersects it in at most d points. De nition 5.2.9. Let n d + 1 be an integer. A cyclic polytope, denoted C (n d ), is the convex hull of any n distinct points on M . The notation C (n d ) is justi ed since, as we shall see in a moment, its face lattice depends only on n and d . We rst observe Proposition 5.2.10. Any d +1 distinct points on M are anely independent. In particular, C (n d ) is a simplicial d-polytope. Proof. Let 0 . . . d be the distinct parameters of these points. We need to show that the vectors x(1) x(0) . . . x(d ) x(0 ) are linearly
;
;
independent, or, equivalently, that the corresponding matrix with these row vectors is non-singular. Clearly, this is the case if and only if the Vandermonde matrix 0 1 0 20 d0 1 B 1 1 21 d1 CC A=B .. @ ... A . 1 d 2d dd Q is non-singular. The determinant of A is known to be 0 i<j d (i ; j ), and this expression is non-zero since the i are pairwise distinct. Next we determine the vertex scheme (C (n d )) which encodes the combinatorial properties of C (n d ). Let C (n d ) be the convex hull of the points xi = x(i ), 1 < 2 < < n , n d + 1. A subset X of V = fx1 . . . xn g will be called an end set if there exists an integer i, 1 i n, such that either X = fx1 . . . xi g or X = fxi . . . xng. The set X will be called contiguous if there exist integers 1 < i j < n such that X = fxi . . . xj g, and an odd (even) contiguous set if it is contiguous and jX j is odd (even). It is clear that any proper subset W V has a unique decomposition W = Y1 X1 X2 Xt Y2
227
5.2. Polytopes
where the Xi are contiguous, and Y1 and Y2 are end sets or empty. The set W is of type (r s) if jW j = r, and if there are exactly s odd contiguous subsets Xi of W . Theorem 5.2.11. Let j be an integer with 0 j d ; 1. A subset W V is a j-face of C (n d ) if and only if W is of type (j + 1 s) for some s with 0 s d ; j ; 1. Proof. We rst show the assertion for j = d ; 1. Since C (n d ) is simplicial, any (d ; 1)-face has d vertices. Thus we have to show that if W V is of type (d s), then conv W is a facet if and only if s = 0. By 5.2.10, the points of W are anely independent, and hence de ne a hyperplane H Rd . It is clear that W H \ M . But actually, W = H \ M since M is a curve of degree d , and it follows that the points of W divide M into d + 1 arcs lying alternately on each side of H . Now conv W is a facet of C (n d ) if and only if H supports C (n d ), or in other words, if and only if all points of V n W lie on one side of H . Obviously this happens exactly when every two points of V n W are separated by an even number of points of W , that is, when s = 0. Let us now treat the general case, and assume that jW j = j + 1. Suppose that W has at most d ; j ; 1 odd contiguous subsets. Then it is possible to nd a subset T of d ; j ; 1 points of M such that V \ T = , and W T has only even contiguous subsets. Since jW T j = d it follows from the rst part of the proof that conv(W T ) is a facet of C (n + d ; j ; 1 d ) supported by the hyperplane H = a (W T ). As W = H \ V we conclude that conv W = H \ C (n d ) is a face of C (n d ). Conversely, if conv W is a j -face of C (n d ), then there exists some facet conv W 0 of C (n d ) with W W 0 . Since W 0 has no odd contiguous subsets, W can have at most d ; j ; 1 odd contiguous subsets. Corollary 5.2.12. The combinatorial type of a cyclic polytope C (n d ) de-
pends only upon n and d, and not on the particular vertex set V M. A polytope P has the highest possible number of j -faces when every subset of j + 1 elements of the vertex set of P is the set of vertices of a proper face of P . In this case we say that P is (j + 1)-neighbourly. Corollary 5.2.13. C (n d ) is d=2]-neighbourly.
The upper bound theorem. In 1957 Motzkin 278] made the following conjecture. Let P be a d -polytope with n vertices then fj (P ) fj (C (n d )) for all j , 1 j d . This conjecture was proved in 1970 by McMullen 271]. We indicate the ideas of his proof: given a d -polytope P with n vertices, one applies in a rst step a process, known as `pulling the vertices', with the e ect of transforming P into a simplicial polytope with the same number of vertices as P , and at least as many faces of
228
5. Stanley{Reisner rings
higher dimension. Thus one may assume from the beginning that P is a simplicial polytope. Just as for simplicial P complexes P one de nes the h-vector (h0 . . . hd ) of P by the equation di=0 hi ti = di=0 fi;1ti (1 ; t)d ;i, f;1 = 1. Then owing to the fact that C (n d ) is d=2]-neighbourly we have ; (a) hi (C (n d )) = n;d +i i;1 for all i, 0 i d=2]. Moreover, the existence of a line shelling of P (see below) yields ; (b) 0 hi(P ) n;d +i i;1 , and (c) hi (P ) = hd ;i (P ) for all i, 0 i d . The identities in (c) are the famous Dehn{Sommerville equations. Now (a), (b) and (c) imply hi (P ) hi (C (n d )) for all i, 0 i d . Finally, since the fj (P ) are non-negative linear combinations of the hi (P ) (see 5.1.8), the proof of the upper bound theorem is completed. Shellings. A shelling of the boundary complex of a d -polytope P (or simply a shelling of P ) is an order of its facets F1 . . . Fm such that S j ;1 Fi i=0 Fi is homeomorphic to a (d 2)-dimensional ball or sphere for all j , 2 j m. Theorem 5.2.14 (Bruggesser{Mani). Every polytope is shellable. We give a sketch of the proof. Present P as an intersection of
\
;
closed half-spaces. One may assume without loss of generality that P = fx 2 Rd : hai xi 1g, 0 i m, where ai is a normal vector for the face Fi . Choose a vector c such that hai ci 6= 0, and order the faces in such a way that ha1 ci > ha2 ci > > ham ci. Then F1 . . . Fm is a shelling of P . Such a shelling is called a line shelling of P . It can be imagined as follows: moving along the line L in direction c starting from the origin, one lists the facets of P as they become `visible'. (This happens exactly when one meets the corresponding supporting hyperplane.) Coming back from the opposite side one lists the remaining facets in the order they `disappear'. Corollary 5.2.15. Let F1 . . . Fm be a line shelling of the polytope P . Then Fm Fm;1 . . . F1 is a line shelling of P , too. Proof. Let F1 . . . Fm be the line shelling induced by c. Then Fm . . . F1 is the line shelling induced by ;c. Suppose now that P is a simplicial polytope. Then the h-vector of P and the h-vector of the vertex scheme (P ) coincide. Furthermore, it is clear that a shelling of P induces a shelling of (P ) (in the sense of Section 5.1). Thus we may apply 5.1.14 to compute the h-vector from a line shelling of P . In particular, it follows from 5.1.14 that hi 0 for all i. Moreover, in view of 5.2.15, we obtain the Dehn{Sommerville equations,
5.3. Local cohomology of Stanley{Reisner rings
229
Theorem 5.2.16 (Sommerville). Let (h0 . . . hd ) be the h-vector of a simplicial polytope. Then hi = hd ;i for 0 i d. These formulas imply in particular that hd = 1. Thus 5.1.9 yields Corollary 5.2.17. Let P be a simplicial d-polytope with f-vector (f0 . . . fd ;1). Then
X
d ;1 i=0
(;1)ifi = 1 ; (;1)d :
This formula is valid not only for simplicial polytopes, but more generally for all polytopes, and is known as; the Euler relation. n;d +i;1 For the proof of the inequalities hi one again uses line i shellings. We refer the reader to McMullen's original paper 271] or 272]. Exercise Let (n d ) denote the boundary complex of the cyclic polytope C (n d ). (a) Show that the cyclic permutation xi 7! xi+1 mod n induces an automorphism of (n d ) for d even. (b) Show that the substitution X1 7! X1 Xn+1 , Xi 7! Xi for i = 2 . . . n, maps the monomial generators of I(nd ) , d even, to those of I(n+1d +1) .
5.2.18.
5.3 Local cohomology of Stanley{Reisner rings We will compute the local cohomology of a Stanley{Reisner ring k] % complex C introduced in Section 3.5. It in terms of the modi ed Cech is not surprising that C , just like k], is equipped with a ne grading. This allows us to decompose the local cohomology groups of k]. As it turns out, their homogeneous pieces can be interpreted as the reduced simplicial homology of certain subcomplexes of . This basic result of Hochster is the main content of this section. As a corollary one obtains Reisner's Cohen{Macaulay criterion for simplicial complexes. For the reader's convenience we recall the notion of reduced simplicial homology. Let be a simplicial complex with vertex set V . An orientation on is a linear order on V . A simplicial complex together with an orientation is an oriented simplicial complex. Suppose is an oriented simplicial complex of dimension d ; 1, and F 2 an i-face. We write F = v0 . . . vi ] if F = fv0 . . . vi g and v0 < v1 < < vi , and F = ] if F = . Having introduced this notation, we de ne the augmented oriented chain complex of , .
.
Ce (): 0 ;! Cd;1 ;!@ Cd;2 ;! ;! C0 ;!@ C;1 ;! 0
230
5. Stanley{Reisner rings
by setting
Ci = M Z F
and
F 2
@F =
dim F =i
Xi
(;1)j Fj
j =0
for all F 2 here Fj = v0 . . . v^j . . . vi ] for F = v0 . . . vi ]. A straightforward computation shows that @ @ = 0. Let G be an Abelian group. We set e i( G) = Hi(Ce () G) i = ;1 . . . d ; 1 H and call He i ( G) the i-th reduced simplicial homology of with values in G. It follows from the next lemma that a reference to the orientation is superuous. Lemma 5.3.1. De ne Ce 0 () in the same way as Ce (), but with respect to a
dierent orientation of . Then there exists an isomorphism of complexes e () = e 0(). Proof. Let < and be the di erent linear orders on the vertex set of V . Given F = v0 . . . vi , v0 < v1 < < vi , there exists a permutation v(i). We leave = F of the vertices of F such that v(0) v(1) it to the reader to verify that : e () e () with (F ) = ( F )F is the
C C
f
g
C !C
desired isomorphism. The i-th reduced simplicial cohomology of with values in G is de ned to be e i( G) = H i(HomZ(Ce () G)) i = ;1 . . . d ; 1: H We set He i () = He i ( Z) and He i () = He i ( Z) for all i. The simplicial complex is called acyclic if He () = 0. In this case, Ce () is split exact, and so He ( G) = 0 and He ( G) = 0 for all Abelian groups G. Examples of acyclic simplicial complexes are the cones: the cone cn() of is the join (see 5.1.20) of a point = fv0 g with . The reader is referred to Exercise 5.3.10 for further details. The cone construction can be iterated. We set cnj () = cn(cnj ;1()) for all j > 1. It is immediate that cnj () is the join of with a j -simplex, and it follows that e (j-simplex ) = 0: (5) H If G = k is a eld, then the reduced simplicial homology and cohomology groups are k-vector spaces, and there are canonical isomorphisms e i( k) = Homk (He i( k) k) He i( k) = Homk(He i( k) k) H for all i see Exercise 5.3.11. In particular it follows that dim He i ( k) = dim He i ( k) for all i. .
.
.
.
5.3. Local cohomology of Stanley{Reisner rings
231
Since Ci k is a vector space of dimension fi, elementary linear algebra yields
X
d ;1
(;1)i dim He i ( k) =
i=;1
X
d ;1
(;1)i fi:
i=;1
This sum, denoted by e () is called the reduced Euler characteristic of . A comparison with the Euler characteristic () introduced in Section 5.1 shows that e () = () ; 1, and we can rewrite the Euler relation 5.2.17 as e () = (;1)d ;1. A geometric realization of in Rn inherits the structure of a topological space (with the subspace topology). In Section 5.2 we denoted this space by jj and remarked that it is unique up to homeomorphism. Let X be a topological space, and ' : jj ! X a homeomorphism. The pair ( ') is called a triangulation of X . Less precisely, we often say in this situation that is a triangulation of X . It is a fundamental theorem in topology (see 280], Theorem 34.3) that the reduced singular homology He i (X k) of a topological space X with triangulation can be computed by means of the reduced simplicial homology of . Theorem 5.3.2. Let X be a topological space with triangulation . Then He i (X k) = He i ( k) for all i.
Examples 5.3.3. (a) Let be the d -simplex with vertices V = fv0 . . . vd g. Then jj is homeomorphic to the d -dimensional closed ball B d , whose reduced singular homology is trivial since B d is contractible to a point. Thus 5.3.2 implies that He ( k) = 0. That the reduced simplicial homology of is trivial can be seen directly: one immediately identi es Ce k with the Koszul complex K (f) associated with f : kd +1 ! k where d +1 f maps the canonical basis elements of k to 1. It follows from 1.6.5(b) that this Koszul complex is exact. (b) Consider the subcomplex ; obtained from by deleting the face F = fv0 . . . vd g then j; j is homeomorphic to the (d ; 1)-dimensional sphere S d ;1. It is clear that the quotient U = Ce ()=Ce (; ) has Ui = 0 for i 6= d and Ud = Z v0 . . . vd ]. Therefore e i(S d;1 k) = He i(; k) = k0 ifif ii =6= dd ;; 1,1. H (c) Let be the vertex scheme of a simplicial (d ; 1)-polytope P . Then jj is homeomorphic to the (d ; 1)-sphere. Therefore, by (b), .
.
.
.
e () =
X
d ;1
(;1)i dim He i(S d ;1 k) = (;1)d ;1:
i=;1
232
5. Stanley{Reisner rings
Thus we have recovered the Euler relation. The following notions will be crucial in the analysis of the local cohomology of a Stanley{Reisner ring. De nition 5.3.4. Let be a simplicial complex, and F a subset of the vertex set of . The star of F is the set st F = fG 2 : F G 2 g, and the link of F is the set lk F = fG : F G 2 F \ G = g. To simplify notation we occasionally omit the index in st or lk . It is clear that st F is a subcomplex of , lk F a subcomplex of st F , and that st F = lk F = if F 62 . In Figure 5.7 let v be the vertex in the centre of the hexagon. Then st v is the full simplicial complex, while lk v is the subcomplex constituting the boundary of the hexagon.
...................................................................................... ... .... . . . . . . . . . . . . . . . . .... ... ... . ... . . . . . . . . . .... .... .... . ..... . . . . . . . . ... . ... ... . . ..... . . . . . . . .... . . ... ..... . . . . . . ...... . . . . . . . . . . . . . ....... . . . . . ...... .. . . . . . . . . . . . ... . . . ... .. . . . . . . . . . ....... . . . . . . . . . . . .... . . . . . . . . . .... ... . . . . . .... . . . . . .. . . . . . . .... ... . . . . . . ... . . . . ... . . . . . . ... ..... . . . . . . . . . . . . . ....... . . . . . . ...... . . . . . . . . . . . . . ..... .. .. . . . .. . . . . . . . . .. .. . . . . . . . . . . . . . . .... . . . . .... . . . . . . . . .... ... . . . . . . . . . ... . .... . . . . . . . . ..... ... . . . . . . . . . . .... ... . . . . . . . . . ... ..................................................................................................................................................................................................................... .... . . . . . . . . . . . . . ... ..... . . . . . . . . . . . . . . . . . ... ... . . . . . . . . . . . . . . . . ... . . .... . . . . . . . . . . . . ... ... . . . . . . . . ..... . . ... . . . . . . . . . ... ... . . . . . . . .... . . ..... . . . . . . . . .. ... . . . . . . . . . . . . . . ... . . . . . . .... . . . . . . . . . . . . . .... ... . . . . . . .. . . . . .... . . . . . . .. ... . . . . . .... . . . . . ... . . . . . . ... ... . . . . .... . . . . . . .. . . . . . ... .... . . . . ... . . . . . . ... . . . . ... .... . . . . . . ..... . . . . . . . . . . . . . .... . . . . . . . .... ... . . ... . . . . . . . . ... . . . ... ... . . .. . . . . . . . . . .. . . ... ..... . .... . . . . . . . . .... . ... ..... ..... . . . . . . . . . . . . . . . . . . . .... . .... ..........................................................................................
Figure 5.7
Lemma 5.3.5. Let F be a face of the simplicial complex , and G 2 lk F. Then (a) F
2 lk G and lkst G F = hGi lklk G F (b) lkst G F is acyclic, if G 6= . Proof. Statement (a) is trivial, and (b) follows from equation (5). Local cohomology. Let be a simplicial complex, k a eld, and R = kX1 . . . Xn]=I the Stanley{Reisner ring of . Let be the maximal ideal generated by the residue classes xi of the indeterminates Xi . Note that (R ) is a local ring, and hence by 2.1.27, R is Cohen{Macaulay if and only if R is Cohen{Macaulay. Thus, in order to determine when is Cohen{Macaulay, we are led to compute the local cohomology H R (R ) of R . To simplify notation we will write H (R ) for H R (R ). Let x = x1 . . . xn as in Section 3.4 we consider the complex ; lim K (xk ) ! which is isomorphic to C : 0 ;! C 0 ;! C 1 ;! ;! C n ;! 0 M Ct = Rxi xi xit m
m
m
.
.
m
m
m
m
m
.
m
m
m
.
.
1 i1 0g: Lemma 5.3.6. (a) dimk (Rx )a 1 for all a 2 Zn. (b) (Rx )a = k if and only if F Ga and F Ha 2 . Proof. (a) Let ri =xni , i = 1 2, be non-zero elements in (Rx )a. Then xn r1 and xn r2 are homogeneous of the same degree, and hence are linearly 1
1
2
1
234
5. Stanley{Reisner rings
dependent over k. We may assume that "(xn r1 ) = xn r2 for some " 2 k then "(r1 =xn ) = r2=xn . (b) We have (Rx )a 6= 0 if and only if there exist a monomial v in R and an integer l such that (i) xmv 6= 0 for all m 2 N and (ii) deg v=xl = a: Condition (i) is equivalent to (i0) v=xl 6= 0. Now (i) implies F supp v 2 , and (ii) implies F Ga and Ha supp v. In particular, F Ha 2 . Q Conversely, suppose F Ga and F Ha 2 . Set v = ai>0 xai i , Q w = ai 1 for some a 2 Zn. Indeed, consider the 1-dimensional simplex in Figure 5.9. By (8), the ne Hilbert series of its canonical v4
v2
v1
.. ... ... .. ... ... .. ......... . . . . . . ...... .... . . ....... . . . .... . ....... . . . . .. ...... .. ........
v3
Figure 5.9
module is
X4 t1 ti 2 t1 + : 1 ; t1 i=2 (1 ; t1)(1 ; ti )
Thus for a = (1 0 0 0) we have dimk (!k])a = 2. Theorem 5.7.1. Let be a (d ; 1)-dimensional Cohen{Macaulay complex
over a eld k. Then the following conditions are equivalent: (a) is not an Euler complex, and there exists an embedding !k] k] of Zn-graded k]-modules (b) there exists a (d 2)-dimensional subcomplex of which is Euler and Cohen{Macaulay over k such that for all F
!
;
0
e dim lk F (lk F k) = k H
2 if F 2 , if F 2= .
If the equivalent conditions hold, then as a Zn-graded k]-module, !k] is isomorphic Q to the ideal J in k] which is generated by the monomials xF = vi 2F xi , F .
2 n
) (b): Let I be the Zn-graded ideal in k] which is isomorphic to the image of !k] ! k]. As we assume that is not an Euler complex, Exercise 5.6.7 implies that I = 6 k]. Further note that if xa 2 I , Proof. (a)
248
5. Stanley{Reisner rings
Q
then ai>0 xi 2 I . This can be deduced from the Hilbert series of !k] see (11). Thus if we set = fsupp u : u 2= I g, then k]=!k] = k]=I = k ]: It follows that is a (d ; 2)-dimensional Cohen{Macaulay complex over k. Since Hk ](t ) = Hk] (t ) ; H!k (t ) we conclude from Exercise 5.6.6 that Hk ] (t;1 1 . . . t;n 1) = (;1)d H!k (t ) ; (;1)d Hk] (t ) = (;1)d ;1(Hk] (t ) ; H!k (t )) = (;1)d ;1Hk ](t ): By 5.6.7, this implies that is an Euler complex. Once again applying (11) we obtain X e Y ti dimk H dim lk F (lk F k) = H!k (t ) = Hk] (t ) ; Hk ] (t ) 1 ; ti F 2 vi 2F ]
]
]
]
=
X Y
ti
1 ; ti F 2n vi 2F
:
A comparison of the coecients on both sides yields the assertion concerning the links of the faces of . Moreover it follows that I equals the ideal J described in the theorem since both ideals have the same Hilbert series. (b) ) (a): First observe that is not an Euler complex, since the links of the faces which belong to are acyclic. In order to obtain the desired embedding of the canonical module we add a vertex w, form the cone cn( ) = fwg , and let ; = cn( ). Then dim ; = dim = d ; 1, k; ] = kX1 . . . Xn Y ]=I; (Y corresponding to the vertex w), and k; ]=(y) = k] where y denotes the residue of Y modulo I; . We will show that ; is an Euler complex which is Cohen{Macaulay over k. In particular ; will be Gorenstein. Then 3.6.12 implies !k] = Homk; ] (k] k; ]) = Ann(y) = Jk; ] = J: Since these isomorphisms are obviously Zn-graded, the desired conclusion follows. It remains to be shown that the links of the faces F 2 ; are homology spheres, that is, satisfy condition (7) of Section 5.4. We distinguish several cases: (i) F 2 n then lk; F = lk F , and (7) is satis ed by assumption. (ii) F 2 then lk; F = cn(lk F ) lk F . Since cn(lk F ) \ lk F = lk F , the Mayer{Vietoris sequence (280], Theorem 25.1) applied to this situation yields the long exact sequence ;! He i(lk F k) ;! He i(cn(lk F ) k) He i(lk F k) ;! He i(lk; F k) ;! He i;1(lk F k) ;!
249
5.7. The canonical module of a Stanley{Reisner ring
provided lk F 6= . Note that He (lk F k) = 0 by assumption, and e (cn(lk F ) k) = 0 by Exercise 5.3.10, so that H .
.
e i;1(lk F k) = He i(lk; F k) H
for all i:
As is an Euler complex which is Cohen{Macaulay over k, it follows that lk; F is a homology sphere. If lk F = , then lk; F = fwg lk F . Note that F is a facet of , so that dim lk F = 0. Hence assumption (b) implies that lk F consists of one vertex. Therefore lk; F = fw vg where v is a vertex of , and thus it is a sphere. (iii) w 2 F then F = fwg G where G 2 , and lk; F = lk G. Again we derive the desired conclusion. Let be a simplicial complex whose geometric realization X = jj is a manifold with boundary @X . Then @X = j j where is the subcomplex of which is characterized by the property that its facets are faces of precisely one facet of (280], x35 and Exercise 4). As an application of 5.7.1 we obtain Theorem 5.7.2 (Hochster). Let k be a eld, and a (d ; 1)-dimensional Cohen{Macaulay complex over k whose geometric realization X = jj is a
manifold with a non-empty boundary @X. Further let be the subcomplex of with @X = , and J the ideal in k] generated by the monomials xF , F . Then the following conditions are equivalent: (a) !k] = J as a Zn -graded k]-module (b) is a Gorenstein complex over k (c) is an Euler complex which is Cohen{Macaulay over k.
2 n
jj
(a) ) (b): Suppose J is the canonical module of k]. Then 3.3.18(b) in conjunction with 3.6.20(c) shows that k ] = k]=J is Gorenstein. (b) ) (c): By 5.6.2, it suces to show that = core . Suppose this is not the case. Then there exists a vertex v 2 such that st v = , and so = fvg ; for some subcomplex ; of . But then @(@X ) = @j j j; j 6= , a contradiction since the boundary of a manifold is a manifold without boundary. (c) ) (a): We have to check the conditions 5.7.1(b) for the links of the faces of . Let : ! Rn be the map de ning the geometric realization of . Suppose F 2 , F 6= , and p 2 relint(conv (F )). If lk F 6= , then 5.4.5 yields Proof.
He dim lk F (lk F k) = Hd ;1 (X X
n fpg k) =
0
2 2
if p @X , k if p = @X .
250
5. Stanley{Reisner rings
The rst case happens when F 2 , the second when F 2= . If lk F = , then F 2= and again He dimlk F (lk F k) = He ;1( k) = k. Now suppose F = . Then lk F = , and we need P to show that eH d;1(X k) = 0, or equivalently, that any (d ; 1)-cycle z = aF F of the chain complex Ce () is trivial. As z is a cycle we have
X
F F 0 dim F =d ;1
aF = 0
for all F 0 2 with dim F 0 = d ; 2. Now since X is a manifold with boundary, each (d ; 2)-face F 0 2 is a face of precisely one facet of when F 0 2 , and of precisely two facets of when F 0 2= . Hence (i) aF = 0 if F contains a facet F 0 2 , and (ii) aF aF = 0 if F 0 2= and F1 and F2 are the facets of containing F 0 . Since by assumption 6= , we conclude from (i) that aF = 0 for at least one facet of . Now let G 2 be any other facet. Notice that is connected since it is Cohen{Macaulay of positive dimension see Exercise 5.1.26. Therefore we can nd a chain of faces F = F0 F1 F2 F2m;1 F2m = G with alternating inclusions where dim F2i = d ; 1 and dim F2i;1 = d ; 2 for i = 0 . . . m. Thus it follows from (ii) and by induction on i that aF i = 0 for i = 0 . . . m in particular, aG = 0. 1
2
2
Zn-graded embedding
of !k] . Though the canonical module of a Stanley{Reisner ring k] cannot always be identi ed with a Zn-graded ideal, it may be realized as a kernel of a certain Zn -graded homomorphism. In order to derive such a presentation we rst observe that the homology of the complex C is concentrated in `negative degrees' see Theorem 5.3.8. To be precise, we have H (Ca) = 0 if some component ai of a is positive. Thus if we set
A
.
.
Di =
M
a2Zn;
.
Cai
where Zn; = fa 2 Zn : ai 0 for i = 1 . . . ng, then H (D ) = H (C ) = H (k]) and these are isomorphisms of Z-graded modules. Write k] = kX ]=I with kX ] = kX1 . . . Xn]. Then by virtue of the local duality theorem for graded modules we obtain the isomorphisms .
.
.
.
.
m
H i (D. )_ = Extnk;Xi ](k] kX ])
i
0
where H i (D )_ = Homk (H i (D ) k) = Hi (Homk (D k)). .
.
.
251
5.7. The canonical module of a Stanley{Reisner ring
Let us more closely inspect the complex G = Homk (D k). Recall that C t is a direct sum of modules Rxi xit where R = k]. Let F = fvi . . . vit g, X = Xi Xit and x = xi xit then Rx = kXi Xi;1 : vi 2 F ]Xi : vi 2= F ]=(I )X where (I )X is an ideal generated by certain squarefree monomials in the variables Xi for which vi 2= F . It is clear that 1 2 (I )X if and only if F 2= . Thus we see that M 0 (Rx)a = kX ;1 : v 2 F ] ifif FF 22= ,, i i a2Zn .
.
1
1
1
L so that Homk (
1
;
(Rx )a k) = kX1 . . . Xn]= F if = kXi : vi 2 F ] F 2 . By de nition, Gt is a direct sum of such modules. Thus we have Theorem 5.7.3. Let be a (d ; 1)-dimensional simplicial complex, and k a eld. For each i = 0 . . . d let Gi be the direct sum of the k]-modules kX1 . . . Xn]= F where F 2 and jF j = i. Consider the complex G : 0 ;! Gd ;! Gd ;1 ;! ;! G1 ;! G0 = k ;! 0 of k]-modules whose dierentiation is composed of the maps (;1)j ;1 nat : kX1 . . . Xn]= F ;! kX1 . . . Xn ]= F 0 if F = fvi . . . vir g and F 0 = fvi . . . bvij . . . vir g, and zero otherwise. Then for i = 0 . . . d, Hi (G ) = Extnk;Xi ](k] kX ]): a2Zn;
P
P
.
P
1
P
1
.
In particular, if is Cohen{Macaulay, then one obtains the exact sequence of Zn-graded k]-modules
0 ;! !k] ;! Gd ;! Gd ;1 ;! ;! G1 ;! G0 ;! 0:
As a consequence of 5.7.3 we derive a result of Grabe 137]. Corollary 5.7.4. Let k be a eld, and a (d ; 1)-dimensional simplicial
complex which is Cohen{Macaulay over k. Then there is a Z-graded embedding !k]( d ) k]:
; ;!
L
Let " : kX1 . . . Xn] ! Gd = jF j=d(kX1 . . . Xn]= F ) be the homomorphism which on Teach component is just the canonical epimorphism. Then Ker " = jF j=d F , and so " induces an isomorphism " : k] ! ImP". Let x = jF j=d xF then x is homogeneous of degree d . Moreover, x is Gd -regular and xGd Im ". To see this, note that if a = (aF ) 2 Gd , then xa = (xF aF ). From this it follows immediately that x is indeed Gd -regular, and it also follows that "(xF ) = xeF for all facets F . Here eF denotes the Proof.
P
P
252
5. Stanley{Reisner rings
element of Gd whose projection to kX1 . . . Xn]= F 0 is 1 if F = F 0 , and 0 otherwise. Since these elements generate Gd , the element x multiplies Gd into the submodule Im ", as asserted. In conclusion we have !k](;d ) = x!k] xGd Im " = k]: We illustrate 5.7.4 by means of the simplicial complex illustrated in Figure 5.9. Theorem 5.7.3 yields the exact sequence P
0 ;! !k] ;!
4 M i=2
kX1 Xi]
;! M kXi] ;! k ;! 0 4
i=1
and it is readily seenLthat !k] is generated by the elements (X1 ;X1 0) P 4 and (X1 0 ;X1) in i=2 kX1 Xi ]. Then ( jF j=2 xF )!k] has the generators (X12X2 ;X12X3 0) and (X12X2 0 ;X12X4 ). Thus we see that the ideal in k] corresponding to x!k] via ";1 is generated by x21 x2 ; x21 x3 and x21x2 ; x21 x4 . Doubly Cohen{Macaulay complexes. Let k be a eld. In Exercise 5.4.9 we noticed that the type rk () of a Cohen{Macaulay complex over k is at least hs , the last non-vanishing component of the h-vector of . Unfortunately, we may have rk () > hs see Exercise 5.7.10. By 5.4.9 equality holds exactly when is level over k. The situation is particularly simple when is level and s = d = dim + 1. Then 5.1.9 implies that rk () = ( 1)d ;1e (), and this number is reasonably accessible. De nition 5.7.5. Let k be a eld. A simplicial complex on the vertex set V is doubly Cohen{Macaulay over k if is Cohen{Macaulay over k, and for all v V the subcomplex V nfvg is Cohen{Macaulay over k of the same dimension as .
;
2
Concluding this chapter we present two results of Baclawski 35] on doubly Cohen{Macaulay complexes. Theorem 5.7.6 (Baclawski). Let k be a eld, and a (d ; 1)-dimensional doubly Cohen{Macaulay complex over k. Then is level and
;
rk () = ( 1)d ;1e ():
We make use of Hochster's formula 5.5.1 which gives the Hilbert series of TorRi (k k]) where R = kX1 . . . Xn]. Note is Cohen{ Macaulay over k if and only if TorRi (k k]) = 0 for i > n ; d . Thus we have the following result: is Cohen{Macaulay over k () He j (W k) = 0 for all W V and j < jW j ; (n ; d ) ; 1:
Proof.
5.7. The canonical module of a Stanley{Reisner ring
253
We claim that TorRn;d (k k])a = 0 for a 6= (1 . . . 1). Suppose this is not the case. Then from 5.5.1 we deduce that there exists a proper subset W of V such that He j (W k) 6= 0 for j = jW j ; (n ; d ) ; 1. Choose i such that W V 0 = V n fvi g then (V 0 )W = W . Since by assumption V 0 is Cohen{Macaulay it follows from (8) (applied to V 0 ) that jW j ; (n ; d ) ; 1 = j jW j ; (n ; 1 ; d ) ; 1, a contradiction. We leave it to the reader to complete the proof. Simply observe that the degrees of the non-zero components of TorRn;d (k k]) determine the degrees of the generators of !k], and that hd = dimk TorRn;d (k k])a for a = (1 . . . 1).
We may view Ce () k as a graded k-vector subspace of k] simply by identifying the elements F 1 with xF for all F 2 . Then He d ;1( k) is identi ed with a k-vector subspace of k]. Corollary 5.7.7 (Baclawski). If is doubly Cohen{Macaulay over k, then as a Z-graded module, !k] (;d ) is isomorphic to the ideal generated by e d;1( k). H Proof. We view !k] as a submodule of Gd . By Exercise 5.7.9, !k] is generated by elements P of degree 0, that is, by elements of Ker((Gd )0 ! (Gd ;1)0 ). Let x = jF j=d xF be as in the proof of 5.7.4. Then x!k] is the ideal in k] which is generated by Ker((xGd )d ! (xGd ;1)d ), and this yields the desired conclusion since (xGd )d ! (xGd ;1)d can be identi ed with Cd ;1() k ! Cd ;2() k. Exercises
Let k be a eld, a simplicial complex, and P a nite k ]-module of rank 1 which is locally free. Show P is free. The proof can be accomplished in the following steps. (a) Let R be a Noetherian ring, P a nite module, and I1 , I2 two ideals in R such that P =Ij P is a free (R=Ij )-module of rank 1 for j = 1 2. Assume that the group of units of R=(I1 \ I2 ) is mapped surjectively onto that of R=(I1 + I2 ). Show P =(I1 \ I2 )P is a free R=(I1 \ I2 )-module of rank 1. (b) Use (a) and induction on the number of facets of . To start the induction observe that a nite, locally free k X1 . . . Xn ]-module of rank 1 is actually free. Indeed such a module is isomorphic to a projective ideal, and since k X1 . . . Xn ] is factorial, projective ideals are principal see 270], Theorem 20.7. 5.7.9. Let k be a eld, and a (d ; 1)-dimensional doubly Cohen{Macaulay complex over k. Show Ker(Gd ! Gd 1 ) is generated by elements of degree 0 in Gd . 5.7.10. Show a 1-dimensional simplicial complex on V satis es r() = e () if and only if for all v 2 V the subcomplex V v is connected. (Reference to a eld k is not needed in dimension 1. Why?) 5.7.8.
;
nf g
254
5. Stanley{Reisner rings
5.7.11. Give an example of a Cohen{Macaulay complex whose type depends on the eld k. 5.7.12. Prove the converse of 5.7.6: if is a Cohen{Macaulay complex and k a eld such that rk () = (;1)d 1 e(), then is doubly Cohen{Macaulay. 5.7.13. Characterize the 1-dimensional simplicial complexes for which there exists a Zn -graded embedding !k ] ! k ]. ;
Notes Simplicial complexes have been considered in topology since Poincare 300] who computed homology groups of topological spaces via triangulations. Another motivation comes from polytope theory where simplicial complexes appear as boundary complexes of simplicial polytopes. The question of how the number of the faces in various dimensions are related to each other has attracted combinatorialists and geometers since Euler who discovered the familiar equation f0 ; f1 + f2 = 2 for 3-polytopes in 1752. A new technique in studying simplicial complexes was introduced by Stanley 356]. His proof of the upper bound theorem for simplicial spheres depends heavily on methods from commutative algebra whose foundations were laid by Hochster 182] and Reisner 306]. Naturally our exposition concentrates on the algebraic aspects of the theory. It is very much inuenced by Stanley's monograph 363] and the lectures by McMullen and Stanley held at the DMV-Seminar in Blaubeuren, July 1991. The reader interested in a general, up-to-date survey on convex polytopes is referred to the excellent article 39] by Bayer and Lee. Hibi's book 169] o ers an attractive introduction to algebraic combinatorics. The results of Kruskal{Katona mentioned in Section 5.1 can be understood as a theorem on Hilbert functions of residue class rings of an exterior algebra see Aramova, Herzog, and Hibi 12] for this approach. Hochster's formula 5.5.1 appeared in 182]. Our treatment is taken from Bruns and Herzog 59] where a more general result for monomial ideals of semigroup rings has been given. There are other notable results in the direction of Baclawski's theorem. For example, Miyazaki 275] proved that the barycentric subdivision of a level complex is again level, and Hibi 166] showed that the proper skeletons of a Cohen{Macaulay complex are all level. There are several aspects in the algebraic theory of simplicial complexes not considered in this book or only discussed in passing: for instance, a careful account of order complexes of posets, or Schenzel's characterization of Buchsbaum complexes see 276], 329], and 365]. It should be mentioned that the statements (i){(v) in Exercise 5.3.15(a) are all equivalent to the Buchsbaum property. Froberg and Hoa 120]
Notes
255
investigated Segre products of Stanley-Reisner rings. For an excellent and up-to-date overview see Stanley 363]. A thorough study of order complexes of posets can be found in Bjorner's paper 45]. Theorem 5.1.12 on the shellability of bounded, semimodular posets is taken from 45]. Garsia's paper 124] is another source of information on this topic. In 165] Hibi classi es those order complexes of distributive lattices which are Gorenstein. In the notes of Chapter 2 we have mentioned the problem as to whether the Poincare series of a local ring is a rational function. For a positively graded ring R over a eld k one de nes its Poincare series with respect to a minimal free graded resolution of k. Froberg 119] showed that if R is de ned by monomial relations of degree 2, then k has a linear resolution over R in particular the Poincare series of R is rational. Backelin 34] proved the rationality of the Poincare series for graded algebras de ned by monomial relations of arbitrary degree. Another important result left out is the g-theorem whose existing proof goes beyond the scope of this book. A vector h = (h0 . . . hd ) 2 Nd+1 satis es the g-condition if h0 = 1, hi = hd;i for all i, and if (h0 h1 ; h0 . . . hd=2] ; hd=2];1) is the h-vector of a homogeneous kalgebra. According to 4.2.10, the latter condition is satis ed if and only if h0 h1 hd=2], and hi+1 ; hi (hi ; hi;1)hii for all i d=2 ; 1. The name g-condition stems from the fact that one commonly denotes by gi the di erences hi ; hi;1. It was conjectured by McMullen in 1971 that (h0 . . . hd ) 2 Nd +1 is the h-vector of a simplicial polytope if and only if it satis es the gcondition. The `suciency' was proved by Billera and Lee 43], while the `necessity' was shown by Stanley 359] who exhibited a homogeneous system of parameters #1 . . . #d of k] such that deg #i = 1 and A = k]=(#1 . . . #d ) has a Lefschetz element, that is, an element ! 2 A1 for which multiplication by ! induces linear maps Ai;1 ! Ai of maximal rank.
6 Semigroup rings and invariant theory
This chapter opens with the study of ane semigroup rings, i.e. subalgebras of Laurent polynomial rings generated by a nite number of monomials. We relate the structure of such a ring R to that of the semigroup C formed by the exponent vectors of the monomials in R , and to the cone D spanned by C . From the face lattice of D we then construct a complex for the local cohomology of R . The connection between R and D is strongest if R is normal: this is the case if and only if R contains all monomials which correspond to the integral points in D. By a theorem of Hochster normal semigroup rings are Cohen{Macaulay. Moreover, we shall determine their canonical modules and, as a combinatorial application, derive the reciprocity laws of Ehrhart and Stanley. We are led to the second topic of this chapter by the fact that rings of invariants of torus actions are normal semigroup rings. We also treat nite groups, covering Watanabe's characterization of Gorenstein invariants and the famous Shephard{Todd theorem on invariants of reection groups. The discussion of invariant theory culminates in the Hochster{ Roberts theorem which warrants the Cohen{Macaulay property for rings of invariants of all linearly reductive groups. 6.1 Ane semigroup rings An ane semigroup C is a nitely generated semigroup which for some n is isomorphic to a subsemigroup of Zn containing 0. Let k be a eld. We write kC ] for the vector space k(C ) , and denote the basis element of kC ] which corresponds to c 2 C by X c. This `monomial' notation is suggested by the fact that kC ] carries a natural multiplication whose table is given by X cX c0 = X c+c0 (we use + to denote the semigroup operation). For example, kZn ] is isomorphic to the Laurent polynomial ring kX1 X1;1 . . . Xn Xn;1] if we let Xi correspond to the i-th element of the canonical basis of Zn similarly kNn ] is isomorphic to kX1 . . . Xn]. The rings kC ] where k is a eld and C is an ane semigroup are called ane semigroup rings. There is a `smallest' group G containing C , characterized by the fact that every homomorphism from C to a group factors in a unique way through G. We write ZC for G, for if C Zn, then G is just the 256
257
6.1. Ane semigroup rings
Z-submodule of Zn n generated by C . Since an ane dsemigroup can be embedded into Z for some n, we see that ZC = Z for some d 2 N which we call the rank of C . We set QC = Q Z ZC and RC = R Z ZC . In the following we will consider ZC as a subgroup of QC and QC as a Q-vector subspace of RC where the inclusions are the map z 7! 1 z and the one induced by the embedding Q ! R. An embedding C ,! Zn of semigroups induces an embedding kC ] ,! kZn ] of k-algebras. Therefore kC ] is a domain it is Noetherian since C is nitely generated. Obviously kC ] and kZC ] have the same eld of fractions if we regard kC ] as a subalgebra of kZC ] in a natural way. It follows that ZC = Zd where d = dim kC ] so dim kC ] = rank C . The ring kC ] is a k-subalgebra of kZC ] it is in fact a graded subring of the ZC -graded ring kZC ] (see Section 5.1 for this notion), and without further speci cation the attributes `graded' and `homogeneous' always refer to the ZC -graduation of kC ]. The graded ideals of kC ] are those
generated by homogeneous elements. Each homogeneous component of kC ] is a one dimensional k-vector space, and therefore the graded ideals correspond to certain subsets of C which will be identi ed below. In order to switch from the ring kC ] to the semigroup C we introduce the operator log I = fc : X c 2 I g for a subset I kC ]: It is clear that log establishes a bijection between the set of graded vector subspaces of kC ] and the set of subsets of C . In a semigroup C we may de ne ideals, and even radical, prime, or primary ideals: S C is an ideal if c + s 2 S for all c 2 C , s 2 S (so is an ideal). The radical of an ideal S is Rad S = fs : ms 2 S for some m 2 Ng Rad S is itself an ideal, and S is a radical ideal if S = Rad S . An ideal S 6= C is prime if c + c0 2 S implies c 2 S or c0 2 S , and it is primary if c + c0 2 S , c 2= S implies c0 2 Rad S . It is easy to check that the radical of a primary ideal is prime. The following proposition whose proof is left for the reader (Exercise 6.1.9) establishes the correspondence of the ideal theory of C and that of the graded ideals of kC ]. Proposition 6.1.1. Let C be an ane semigroup, and I, I 0 kC ] graded
k-vector subspaces. Then (a) I I 0 log I log I 0 , log(I1 I2 ) = log I1 log I2, log I1 + I2 = log I1 log I2 , (b) I is a (radical, prime, primary) ideal if and only if log I is a (radical, prime, primary) ideal furthermore log Rad I = Rad log I, (c) the minimal prime overideals of I are graded.
()
\
\
Normal semigroup rings. An ane semigroup C is called normal if it satis es the following condition: if mz C for some z ZC and m N,
2
2
2
258
6. Semigroup rings and invariant theory
2
m > 0, then z C . One sees immediately that C must be normal if kC ] is a normal domain: X z is an element of the eld of fractions of kC ], and if (X z )m kC ] and kC ] is normal, then X z kC ]. That the
2
2
converse is also true will be shown below. First we explore the geometric signi cance of normal semigroups. A non-empty subset D of an R-vector space V is called a cone if it is closed under linear combinations with non-negative coecients in R. For S V the set
R+ S = f
Xn i=1
ai vi : ai
2 R+ vi 2 S n 2 Ng
is obviously the smallest cone containing S it is the cone generated by S . Finitely generated cones can be characterized in complete analogy with convex polytopes: a subset D of a nite dimensional R-vector space V is a nitely generated cone if and only if there exist nitely many vector half-spaces Hi+ = fv 2 V : hai vi 0g ai 2 V ai 6= 0 i = 1 . . . m + + such that D = H1 \ \ Hm . In the following it will be necessary to consider rational polytopes and cones. Let V be an R-vector space of nite dimension, and U a Q-vector subspace of V such that dimQ U = dimR V . A polytope P V is rational (with respect to U ) if its vertices lie in U , and a cone is rational if it is generated by a subset of U . We choose a scalar product which has an orthonormal basis in U , and de ne a rational half-space to be a set H + = fv 2 V : ha vi g with a 2 U , a 6= 0 and 2 Q. Of course, the notion of rationality makes sense only with respect to a xed Q-subspace U (and, for a half-space, is independent of the choice of the scalar product, provided it has an orthonormal basis in U ). If V = Rn , then it is tacitly understood that U = Qn, and when V = RC for an ane semigroup C , U = QC . We need some results about rational polytopes and cones: (i) A subset P V is a rational polytope if and only if it is bounded and the intersection of nitely many rational half-spaces. (ii) A subset D V is a nitely generated rational cone if and only if it is the intersection of nitely many rational vector half-spaces. (iii) Let v1 . . . vm 2 U . ThenPu 2 U \ convfv1 . . . vm g ifPand only if there exist r1 . . . rm 2 Q+ with mi=1 ri = 1 such that u = mi=1 ri vi in other words U \ convfv1 . . . vm g = convQ fv1 . . . vm g: (iv) Let v1 . . . vm 2 U . Then u 2 UP\ R+fv1 . . . vm g if and only if there exist r1 . . . rm 2 Q+ such that u = mi=1 ri vi in other words U \ R+fv1 . . . vm g = Q+fv1 . . . vm g:
259
6.1. Ane semigroup rings
It is a good exercise for the reader to prove (i){(iv). An essential argument is that a linear system of equations with rational coecients is soluble over Q if and only if it has a solution over R. Proposition 6.1.2 (Gordan's lemma). (a) If C is a normal semigroup, then C = ZC \ R+C (within RC ). (b) Let G be a nitely generated subgroup of Qn and D a nitely generated rational cone in Rn . Then C = G \ D is a normal semigroup. Proof. (a) It follows from (iv) above that ZC \ R+C = ZC \ Q+C , and that C = ZC \ Q+C is (almost) the de nition of a normal semigroup. (b) The essential point to prove is that G \ D is a nitely generated rational semigroup the rest is again elementary. We claim that T D \ RC is a nitely generated rational cone in RC . In fact, let D = Hi+ be given as the intersection of nitely many T + n rational half-spaces of R . Then D \ RC = (Hi \ RC ), and because of QC = RC \ Qn, each Hi+ \ RC is a rational half-space of RC or equal to RC . Replacing G by ZC and Rn by RC we may now assume that G = Zn . P v n By hypothesis there exist q1 . . . qv 2 Q with D = f i=1 ai qi : ai 2 R ai 0g. Multiplying by a suitable common denominator we may assume that q1 . . . qv 2 Zn. P Choose c 2 C . Then c = vi=1 ai qi with ai 2 Q+, and therefore c=
Xv i=1
a0i qi +
Xv i=1
a00i qi
with a0i 2 N and a00i 2 Q, 0 a00i < 1. Since C = P Znv \ D00 , we have00 c00 = P v 00 00 a qi 2 C . But c lies in the bounded set B = f a qi : 0 a < 1g so that Zn \ B is nite. The nite set (B \ Zn) fq1 . . . qv g generates C. The invertible elements in a semigroup C form a group C0, the largest group contained in C . If C0 = 0, we say that C is positive. If C is normal, then C splits into a direct sum of C0 and a positive normal semigroup: Proposition 6.1.3. Let C be a normal semigroup, and C0 the group of its i=1 i
i=1 i
i
invertible elements. (a) Then C = C0 C 0 with a positive normal semigroup C 0. Furthermore C0 = Zu for some u 0. (b) One has kC ] = kC0 ] k kC 0 ] = kZu ] k kC 0 ] for every eld k.
Proof.
It follows immediately from the normality of C that the group
ZC=C0 is torsion-free. Therefore C0 is a direct summand of ZC , and
hence of C itself. The rest of (a) is quite obvious. Part (b) is a special case of the general fact that kC1 C2] = kC1 ] k kC2 ].
260
6. Semigroup rings and invariant theory
With the notation of the previous proposition, all essential ringtheoretic properties are shared by kC ] and kC 0 ]: the ring kC ] arises from kC 0 ] by a polynomial extension followed by the inversion of the indeterminates, and is a free, thus faithfully at, kC 0 ]-module. Theorem 6.1.4. Let C be an ane semigroup, and k a eld. Then the following are equivalent: (a) C is a normal semigroup (b) kC ] is normal.
The implication (b) ) (a) has already been observed. For (a) ) (b) we note that C is the intersection of nitely many rational half-spaces Hi+ = fq 2 RC : hai qi 0g of RC with ZC , ai 2 QC see 6.1.2. Set Ci = ZC \ Hi+. One has (Ci)0 = fz 2 Ci : hai zi = 0g. It follows that (Ci)0 = Zd ;1 where d = rank C . Thus the semigroup Ci0 in the splitting Ci = (Ci)0 Ci0 has rank 1. Since Ci is normal, Ci0 is also normal. Being a normal subsemigroup of Z, and not a group, Ci0 is isomorphic to N. Therefore kCi ] = kZd;1 N] is even regular. As kC ] is the intersection of the normal rings kCi ], it is normal itself. In order to use the results on Z-graded rings and modules for ane semigroup rings we say that a decomposition Proof.
k C ] =
M i2N
kC ]i
of the k-vector space kC ] is an admissible grading if kC ] is a positively graded k-algebra with respect to this decomposition, and furthermore each component kC ]i is a direct sum of nitely many ZC -graded components. It follows that X c is homogeneous for each c 2 C , and that the maximal ideal of kC ] is generated by the monomials X c, c 6= 0. Thus kC ] has an admissible grading only if C is positive. That the converse is also true, will be very important in the following. Proposition 6.1.5. Let C be a positive ane semigroup. Then C is isomorphic with a subsemigroup of Nm for some m. In particular kC ] is isomorphic with a graded k-subalgebra of kX1 . . . Xm], and has an admissible m
grading.
Proof. We choose a scalar product that has a Z-basis of ZC as an orthonormal basis. The cone R+C is the intersection of half-spaces
f 2 RC : hai vi 0g
Hi+ = v
ai
2 QC
6
ai = 0 i = 1 . . . m:
Multiplying by a suitable common denominator we may assume that ai 2 ZC . Then hai ci 2 Z for all c 2 ZC , and ' : ZC ! Zm , '(c) =
6.1. Ane semigroup rings
261
(ha1 ci . . . ham ci) is a group homomorphism with '(C ) Nm . The kernel of ' is the intersection of the hyperplanes Hi = fv 2 RC : hai vi = 0g therefore the group Ker ' \ ZC is contained in C . Since C is positive, 'jC is injective. The rest is obvious. The graded prime ideals of an ane semigroup ring. The results of Sections
6.2 and 6.3 depend crucially on the fact that one can determine the graded prime ideals of kC ] from the geometry of the cone R+C . Let us rst show that the set of non-zero graded radical ideals in kC ] has a unique minimal element. For an ane semigroup C we set relint C = C \ relint R+C: Lemma 6.1.6. Let C be an ane semigroup. Then the ideal generated by the elements X c, c 2 relint C is a radical ideal, and is contained in every non-zero graded radical ideal of kC ]. Proof. In view of 6.1.1 we may equivalently prove that relint C is the smallest non-empty radical ideal of C . Set I = relint C . It is obvious that I is a radical ideal of C . Let J C be an arbitrary non-empty radical ideal, c 2 I , and s 2 J . We must show that c 2 J , for which there is only something to prove if c 6= s. As c 2 relint R+C , the intersection of relint R+C with the line L through s and c is a neighbourhood of c in L. Since L is rational, there exist rational points on both sides of c in L arbitrarily close to c. So there exists t 2 L \ (relint R+C ) \ QC such that c lies in the line segment s t]. Therefore we have an equation c = s + (1 ; )t with 2 Q 0 < < 1: Multiplication with a suitable common denominator yields an equation mc = ns + t0 with m n 2 N nf0g and t0 2 C . It follows that c 2 J because J is a radical ideal and s 2 J . We shall see in Theorem 6.3.5 that the ideal considered in 6.1.6 is the canonical module of kC ] if C is a normal semigroup. Let C be an ane semigroup, and suppose that F is a face of R+C . The set C n F is immediately seen to be a prime ideal of C . By 6.1.1 it follows that the ideal (F ) of kC ] generated by the elements X c, c 2 C n F , is a graded prime ideal of kC ]. In fact, all homogeneous prime ideals can be represented in this way: Theorem 6.1.7. Let C be an ane semigroup, and k a eld. Then the assignment F 7! (F ) is a bijection between the set of non-empty faces of R+C and the set of graded prime ideals of kC ]. P
P
262 Proof.
6. Semigroup rings and invariant theory
In view of 6.1.1 we may equivalently show that the assignment
7! (F ) = C n F is a bijection between the set of non-empty faces of R+C and the set of prime ideals of C . It is easy to see that is injective in fact, F = R+(C \ F ) = R+(C n (F )) for every face F of R+C . Surjectivity of is proved by induction on rank C , the case rank C = 0 being trivial. Let rank C > 0 and P C be a prime ideal. If P = , then P = (R+C ). So suppose P = 6 . By 6.1.6 we have P relint C . As relint C = (F1 ) \ \ (Fm ) where F1 . . . Fm are the maximal proper faces of R+C , it follows that P (Fi ) for at least one i, say P (F1 ). The intersection C \ F1 is an ane semigroup with rank C \ F1 < rank C . As P \ F1 is a prime ideal in C \ F1 , there exists a face G of R+F1 with P \ F1 = (C \ F1) n G. Being a face of a face of R+F1, G is a F
face of R+C , and elementary set theory shows that P = (G). In the next section the homogeneous localizations kC ]( ) will play a crucial role. Since we shall argue rather geometrically, it is more suggestive to denote them by kC ]F where F is the face of R+C with = (F ). This notation is also justi ed by the fact that kC ]F is the ring of fractions of kC ] with respect to the multiplicatively closed set fX c : c 2 C \ F g. Finally we want to relate the faces of the cone R+C to those of a suitably chosen polytope. For simplicity we restrict ourselves to the case in which C is positive. More generally, let D = fx 2 Rn : hai xi 0 for i = 1 . . . mg be a cone in Rn given as the intersection of vector half-spaces de ned by ai 2 Rn , i = 1 . . . m. Let us say that D is positive if 0 is the only element v 2 D with ;v 2 D. This is the case if and only if a1 . . . am generate Rn . Set b = a1 + +am and de ne T = fx 2 D : hb xi = 1g: It follows easily that T is bounded. Being the intersection of nitely many ane half-spaces, it is a convex polytope. We say that the hyperplane fx : hb xi = 1g is transversal to D, and call T a cross-section of D. Crosssections are introduced because their combinatorial structure will lead us to a complex by which one can compute the local cohomology of an ane semigroup ring. A non-empty face of D is given by D itself or by H \ D where H is a supporting hyperplane of D. Since D is a cone, H must contain 0. Therefore there is a unique minimal non-empty face of D, namely f0g, and we choose F(D) to be the set of non-empty faces of D. p
p
P
263
6.2. Local cohomology of ane semigroup rings
Proposition 6.1.8. Let D be a positive cone, and T a cross-section of D. Then the assignment F 7! F \ T induces an isomorphism F(D) = F(T ) of partially ordered sets. Its inverse is given by G
7! R+G.
The proof is easy and left as an exercise for the reader. At several places below we will have to use the correspondence between the faces of R+C and those of a cross-section T of R+C , as given by 6.1.8. In order to avoid cumbersome notation we agree on denoting corresponding faces by corresponding capital and small letters. So, if F is a face of R+C , then f = (R+C ) \ T . Exercises Prove Proposition 6.1.1. Hint: For the implication `(' in (b) and for (c) one uses that ZC
= Zd , d = rank C , can be given a linear order under which it becomes an ordered group. (For example one may choose the reverse degree-lexicographical order introduced in Section 4.2.) Then the homogeneous components of an element are linearly ordered, and one argues similarly as in the proof of Lemma 1.5.6. 6.1.10. Let S , T be a ne semigroups, S T . One says that S is a full subsemigroup of T if S = T \ ZS . Show (a) a full subsemigroup of a normal semigroup is again normal, (b) a positive a ne semigroup is normal if and only if it is isomorphic to a full subsemigroup of Nn for some n 0, (c) if S is full in T , then k S ] is a direct k S ]-summand of k T ]. 6.1.11. Let C be an a ne semigroup. Then k C ] is regular if and only if C is of the form Zu Nv . Hint: The implication `(' is easy. For the implication `)' one uses 6.1.5 and 2.2.25, noting that a minimal set of generators of the maximal ideal of k C ] can be chosen of the form X c . . . X cv . 6.1.12. Let C be an a ne semigroup, and F a face of R+C . Show (a) the composition k C \ F ] ! k C ] ! k C ]= (F ) of natural maps is an isomorphism of a ne semigroup rings, (b) if C is normal, then k C \ F ] is also normal, (c) k C ]F is an a ne semigroup ring. 6.1.13. Let D Rn be a positive cone, and z 2 Rn . Show that z 2 = ;D if and only if there exists a hyperplane H which is transversal to D and contains z . 6.1.9.
1
P
6.2 Local cohomology of ane semigroup rings In this section we shall de ne a complex by which we can compute the local cohomology of an ane semigroup ring it is based on a construction of algebraic topology, namely the oriented augmented chain complex associated with a nite regular cell complex.
264
6. Semigroup rings and invariant theory
Cell complexes. Regular cell complexes generalize the simplicial com-
plexes of Chapter 5. Massey 266] gives an introduction to the theory of cell complexes which is very well suited for our purpose. We introduce the chain complex associated with a cell complex axiomatically, borrowing the existence and uniqueness theorems from algebraic topology. A nite regular cell complex is a non-empty topological space X together with a nite set ; of subsets of X such that the following conditions are satis ed: S (i) X = e2; e (ii) the subsets e 2 ; are pairwise disjoint (iii) for each e 2 ; , e 6= there exists a homeomorphism from a closed i-dimensional ball B i = fx 2 Ri : kxk 1g onto the closure e of e which maps the open ball U i = fx 2 Ri : kxk < 1g onto e (iv) 2 ; . By the invariance of dimension the number i in (iii) is uniquely determined by e, and e is called an open i-cell is a (;1)-cell. By ; i we denote the set of the i-cells in ; . The dimension of ; is given by dim ; = maxfi : ; i 6= g. It is nite since ; is nite. One sets j; j = X . Finite regular cell complexes are special cases of a more general topological structure, namely that of a CW-complex. Since all our CWcomplexes are nite and regular, we shall simply call them cell complexes. A cell e0 is a face of the cell e 6= e0 if e0 e, and a subset of ; is a subcomplex if for each e 2 all the faces of e are contained in . The classical examples of cell complexes S are convex polytopes P together with their decomposition P = f2F(P ) relint f . For them the following property, which follows from (i){(iv), is an elementary theorem: (v) if e 2 ; i and e0 2 ; i;2 is a face of e, then there exist exactly two cells e1, e2 2 ; i;1 such that ej is a face of e and e0 is a face of ej . Each simplicial complex may be identi ed with a cell complex, namely the cell complex it de nes in a natural way on a geometric realization and whose open cells correspond to the faces of . It is convenient to denote this cell complex simply by , and an open cell by the corresponding face of . Let fv1 . . . vn g be the vertex set of . For e 2 i and e0 2 i;1 we set "(e e0 ) = 0 if e0 is not a face of e, and "(e e0 ) = (;1)k+1 if e corresponds to fvi . . . vim g and e0 to fvi . . . bvik . . . vim g, i1 < < im . Then the augmented oriented chain complex of , which has L been introduced in Chapter 5, is a complex of i () = free Z -modules C i Ze whose di erential is given by @(e) = P 0 i; "(e e0)e0. The crucialc2point in constructing a similar complex for e 2 an arbitrary cell complex is to nd a suitable function ". Let us say that " is an incidence function on ; if the following conditions are satis ed: 1
1
1
265
6.2. Local cohomology of ane semigroup rings
(a) to each pair (e e0 ) such that e 2 ; i and e0 2 ; i;1 for some i 0, " assigns a number "(e e0 ) 2 f0 1g (b) "(e e0 ) 6= 0 () e0 is a face of e (c) "(e ) = 1 for all 0-cells e (d) if e 2 ; i and e0 2 ; i;2 is aface of e, then "(e e1)"(e1 e0 ) + "(e e2)"(e2 e0 ) = 0 where e1 and e2 are those (i ; 1)-cells such that ej is a face of e and e0 is a face of ej (see (v) above). Lemma 6.2.1. Let ; be a cell complex. Then there exists an incidence function on ; .
For a proof see Lemma IV.7.1 in 266] where the incidence numbers
"(e e0 ) appear as topological data determined by orientations of the cells.
Figure 6.1 indicates two incidence functions on the solid rectangle and how they are induced by orientations.
+ +
+
;
+ "
..................................................................................................................................................... ... .... ... ... ..... .... .. .. ... ... ... .... ... .. ... .... .. .... .. ... .. ... ... .........................................................................................................................................
; .........
# ;
+
n
/
+
+
! + ;
+ ;
+
;
;
................................................................................................................................................... ... .. ... .. ... .... ... .. ... ... ... ... ... ... ... ... ... .. ... .. ... ... ... ... ....................................................................................................................................................
; ........
# +
Figure 6.1
;
n
.
+
+
#+ ;
Let : ; ! f1g be a function with () = 1 and (e) = 1 for all 0-cells e. Then the function "0 (e e0 ) = (e0 )"(e e0 ) (e) is also an incidence function. On the other hand, all pairs ", "0 of incidence functions di er only by a `sign' : Theorem 6.2.2. Let ; be a cell complex with incidence functions " and "0. Then there exists : ; ! f1g such that () = 1 and "0 (e e0 ) =
(e0)"(e e0 ) (e) for all e 2 ; i, e0 2 ; i;1, i = 0 . . . dim ; . This is Theorem IV.7.2 of 266] (in a di erent formulation). Its proof shows that incidence functions can be constructed in a completely naive manner. (i) One starts with ; 0 on which there is no choice according to property (c) of incidence functions. (ii) If one has constructed an incidence function "i on ; 0 ; i , then there exists an incidence function "i+1 on ; 0 ; i+1 whose restriction to ; 0 ; i is just "i. The reader is advised to construct incidence functions for some three dimensional polytopes.
266
6. Semigroup rings and invariant theory
Let ; be a cell complex of dimension d ; 1, and " an incidence function on ; (as in Chapter 5 it is convenient to denote dimension by d ; 1). We de ne the augmented oriented chain complex of ; by the complex
Ce (; ): 0 ;! Cd;1 ;!@ Cd;2 ;! ;! C0 ;!@ C;1 ;! 0 where we set C i = M Ze
and
e2; i
@(e) =
X e02; i;1
"(e e0 )e0 for e
2 ; i
; 1. That @2 = 0 follows from the de nition of an incidence function and property (v) of cell complexes. The notation Ce (; ) is justi ed since the dependence of Ce (; ) on " is inessential Theorem 6.2.2 guarantees that we obtain an isomorphic complex upon replacing " by another incidence function "0 . (The isomorphism is given by e 7! (e)e.) For simplicity of notation we set He i(; ) = Hi (Ce (; )). The fundamental importance of Ce (; ) in algebraic topology relies on i = 0 . . . d
the fact that it computes reduced singular homology : Theorem 6.2.3. Let ; be a cell complex. Then He i (; ) = He i (j; j) for all i 0 (and He ;1 (; ) = 0). Theorem IV.4.2 of 266] states that Hi (C(; )) = Hi (j; j) for the none augmented complex C(; ) which arises from C(; ) if we replace Ce ;1 by 0. It follows easily that He 0(; ) = He 0(j; j) as well. We use 6.2.3 via the following corollary: Corollary 6.2.4. Let ; be a cell complex such that j; j is homeomorphic to e i(; ) = 0 for all i ;1. a closed ball B n. Then H
Local cohomology. Let C be a positive ane semigroup, and k a eld. The ideal in R = kC ] generated by the elements X c, c C 0 , is maximal. For an R -module M we denote by H i (M ) the i-th right derived
2 nf g
m
functor of
m
f 2 M:
= 0 for i 0g: As in 3.5.3 one has a natural isomorphism ; (M ) = x m
;!
ix
m
for all i 0:
H i (M ) = lim ExtiR (R= j M ) m
m
The natural map ExtiR (R= j M ) ! ExtiR (R =( R )j M ) is an isomorphism. Therefore H i (M ) = H i R (M ), and we are justi ed in calling H i (M ) a local cohomology module. We now want to construct a complex m
m
m
m
m
m
m
m
m
m
m
267
6.2. Local cohomology of ane semigroup rings
`computing' H i (M ) which resembles the combinatorial structure of C as closely as possible. Suppose for the moment that C = Nn so that R = kC ] = kX1 . . . Xn ] % complex and = (X1 . . . Xn). As we saw in 3.5.6, the modi ed Cech C : 0 ;! C 0 ;! C 1 ;! ;! C n ;! 0 with M Ct = Rxi xit m
m
.
computes H (M ) in the sense that (M ) = H i (M C ) for all i 0. The t components of C are of the form RF where F is a face of Rn+ = R+Nn, and the di erential is composed of maps " nat: Rxi xit ;! Rxi xit xj whose signs " are just the values of an incidence function on the pair (convfei . . . eit ej g convfei . . . eit g) of faces of the simplex spanned by the canonical basis e1 . . . en of Rn . This simplex is a cross-section of the cone Rn+. It is easy to generalize this construction. Let C be a positive ane semigroup of rank d , R = kC ], T a cross-section of the cone R+C , and F = F(T ) its face lattice. (We remind the reader of our convention of denoting corresponding faces of R+C and T by F and f respectively.) Let M Lt = RF t = 0 . . . d 1
Hi
.
m
.
m
1
1
1
1
f2Ft;1
and de ne @ : Lt;1 ! Lt by specifying its component 0 if F 0 6 F , @f0f : RF 0 ! RF to be 0 "(f f ) nat if F 0 F here " is an incidence function on F. It is clear that L. : 0
;! L0 ;!@ L1 ;! ;! Ld;1 ;!@ Ld ;! 0
is a complex. Theorem 6.2.5. Let C be a positive ane semigroup, and k a eld. Let be the maximal ideal generated by the elements X c, c 2 C n f0g. Then for every kC ]-module M, and all i 0, H i (M ) = H i (L M ): m
.
m
Proof. We follow the pattern of the proof of 3.5.6. Let I
be the ideal generated by the elements X c, c 6= 0 for which there exists a one dimensional face F of R+C with c 2 F . In order to show H 0 (L0 M ) = H 0 (M ) for m
268
6. Semigroup rings and invariant theory
all kC ]-modules M , we must verify that Rad I = . Let c1 . . . cm 2 C be a minimal set of generators of Q+C . Then each one dimensional face F of R+C contains exactly one of the ci , and it is enough to show that Rad J = where J is the ideal generated by X c . . . X cm . Let c 2 C , c 6= 0. There exist q1 . . . qm 2 Q+ with c = q1c1 + + qm cm . Multiplication by a common denominator yields rc = s1c1 + + sm cm with r, si 2 N. Since si 6= 0 for at least one i, it follows that (X c)r 2 J . Now let 0 ! M1 ! M2 ! M3 ! 0 be an exact sequence of kC ]modules. Since all the summands of L are at kC ]-modules, this yields an exact sequence 0 ! L M1 ! L M2 ! L M3 ! 0: As desired we have a long exact sequence ! H i(L M1) ! H i(L M2) ! H i(L M3) ! H i+1(L M1) ! Finally we must show that H i (L M ) = 0 for all i if M is an injective kC ]-module. It suces to consider the indecomposable modules E (R= ) where is a prime ideal of R = kC ]. Then, as shown in the proof of 3.5.6, there are only two possibilities for an element x of R : either every element of E (R= ) is annihilated by some power of x, namely if x 2 , or multiplication by x is bijective on E (R= ). So if F \ 6= , E (R= ) RF = 0 E (R= ) if F \ = . Set P = log . Then P is a prime ideal in the semigroup C , and by 6.1.7 there is a face G of R+C with P = C n G. Thus 0 if F 6 G, E (R= ) RF = E (R= ) if F G. Let G = F(g) denote the face lattice of the face g = G \ T of a cross-section T of R+C . It follows that M E (R= ) Lt E (R= ) = m
1
m
.
.
.
.
.
.
.
.
.
p
p
p
p
p
p
p
p
p
p
p
p
p
p
f2Gt;1
for all t 0. Of course G is a subcomplex of F = F(T ), and the restriction of an incidence function on F to G is an incidence function on G. Therefore we have ; L E (R= ) = HomZ Ce (G)(;1) E (R= ) : (This statement is the heart of the proof the reader should verify it carefully.) Since g is a convex polytope, it is homeomorphic to a closed ball. So Ce (G) is an exact complex see 6.2.4. Since Ce (G) is a complex of free Z-modules, exactness is preserved in HomZ (Ce (G)(;1) E (R= )). .
p
p
p
269
6.2. Local cohomology of ane semigroup rings
Corollary 6.2.6. Let C be a positive ane semigroup of rank d, and k be a eld. Then kC ] is Cohen{Macaulay if and only if H i (L. ) = 0 for i = 0 . . . d 1.
;
Set R = kC ], and note that d = dim R (why?). If R is Cohen{ Macaulay, then R is Cohen{Macaulay. Thus it follows from 3.5.7 that H i (L ) = 0 for i = 0 . . . d ; 1. Conversely, 3.5.7 also implies that R is Cohen{Macaulay if H i (L ) = 0 for i = 0 . . . d ; 1. By virtue of 6.1.5 R is a local ring with maximal ideal . Now 2.1.27 yields that R is Cohen{Macaulay. Proof.
m
m
.
m
.
m
Exercises We will see in the next section that a normal semigroup ring is Cohen{ Macaulay. This exercise presents an example (due to Hochster 174]) of an a ne semigroup ring showing that Serre's condition (S2 ) alone is not su cient for the Cohen{Macaulay property. Let k be a eld, and Y1 , Y2 , Z1 , Z2 be indeterminates over k. Prove: (a) The semigroup C generated by the monomials xij = Yi Zj , i j = 1 2, is normal S = k C ] is a normal domain of dimension 3. (b) The substitution Xij 7! xij induces an isomorphism 6.2.7.
k X11 X12 X21 X22 ]=(X11 X22
; X21 X12 ) = S
S is a Cohen{Macaulay ring. (c) The subsemigroup C of C generated by all monomials f with degY f > 1 and degY f > 1 is nitely generated. (d) The elements x211 , x222 , x212 + x221 form a homogeneous system of parameters of R = k C ], but not an R -sequence. (e) The ideals generated by x211 and x222 in R are unmixed. (Hint: Use that the associated primes of a ZC -graded module are ZC -graded this follows as in 1.5.6.) (f) R x112 x222 ] = S x112 x222 ]. (g) R satis es Serre's condition (S2 ), but is not Cohen{Macaulay. 6.2.8. One says that an n-dimensional positive cone D is simplicial if it is generated by n elements, and a positive a ne semigroup C is simplicial if the cone R+C is simplicial. Let k be a eld. (a) Let C be an arbitrary positive a ne semigroup. Prove that X c . . . X cn with c1 . . . cn 2 C form an k C ]-sequence if and only if X ci X cj is an k C ]-sequence for all i 6= j , equivalently, ci + s = cj + t for s t 2 C implies s 2 ci + C . (b) Show that C is simplicial if and only if k C ] has a homogeneous system of parameters X c . . . X cn with c1 . . . cn 2 C . (c) Let C be simplicial. Deduce from (a) and (b) that k C ] is Cohen{Macaulay if and only if it satis es Serre's condition (S2 ), and that this property is independent of k (Goto, Watanabe, and Suzuki 133]). (d) Formulate a Gorenstein criterion for k C ] with C simplicial, using the socle of k C ]=(X c . . . X cn ), and show that this property is also independent of k. 0
1
2
0
0
;
;
;
0
;
1
1
1
270
6. Semigroup rings and invariant theory
6.3 Normal semigroup rings In this section we want to show that a normal semigroup ring is a Cohen{Macaulay ring and to determine its canonical module. The complex L constructed in the previous section is ZC -graded in a natural way, and in order to compute its cohomology we analyze its graded components just as in the proof of 5.3.8. Given z 2 ZC , the main point is to determine those faces F of C for which (RF )z 6= 0. As we shall see, this is the case if and only if the face F is not `visible' from z . Let P be a polyhedron in a R-vector space V . Let x, y 2 V . We say that y is visible from x if y 6= x and the line segment x y] does not contain a point y0 2 P , y0 6= y. A subset S V is visible if each v 2 S is visible. Proposition 6.3.1. Let P be a polytope in Rn with face lattice F, and x 2 Rn a point outside P . Set S = fF 2 F : FSvisible from xg. Then S is a subcomplex of F its underlying space S = F 2S F is the set of points .
2 P which are visible from x, and is homeomorphic to a closed ball. Proof. Let y 2 P be visible from x. There exists a (unique) face F with y 2 relint F , and one concludes easily (for example by 6.3.2 below) that the whole of F is visible from x. Therefore S = fy 2 P : y visible from xg, and it follows easily that S is homeomorphic to a closed ball. That S is y
a subcomplex is obvious. Let P be a polyhedron in an R-vector space V , dim V < 1. Suppose that P is given as the intersection of nitely many half-spaces
f 2 V : hai xi ig
Hi+ = x
We set
f h i
g
f h i
i = 1 . . . m:
g
f h i g Lemma 6.3.2. With the notation introduced, a point y 2 P is visible from 6 . x 2 V n P if and only if y0 \ x; = x0 = i : ai x = i x+ = i : ai x > i x; = i : ai x < i :
The elementary proof is left for the reader. Figure 6.2 illustrates the following lemma. Let C = N2 R2 , and F be the positive X -axis, G the positive Y -axis. Then kC ]F = kX Y X ;1], and (kC ]F )z 6= 0 for z 2= C exactly when z is in the second quadrant (including the negative X -axis). Thus (kC ]F )z 6= 0 if and only if F is not visible from z . Similar arguments work for the faces f0g, G, and C . Lemma 6.3.3. Let C be a normal semigroup, k a eld, and R = kC ]. Let F be a face of R+C and z 2 ZC. Then (RF )z 6= 0 (and therefore (RF )z = k) if and only if F is not visible from z.
271
6.3. Normal semigroup rings G
C F
Figure 6.2 Proof. Suppose rst that F is not visible from z . Then there exists c C F which is not visible from z . We have c+ z ; (note that c; = ), and it follows that (mc + z ); = for m 0, whence mc + z C . (Of course z +, c; etc. are de ned with respect to a representation of R+C as an intersection of vector half-spaces.) That mc + z C is equivalent with (X c)m X z R so that X z RF . Conversely suppose that (RF )z = 0. Then there exists c C F with X cX z R . Consequently c + z C , and (c + z ); = , which is only possible if c is not visible from z .
2 \
2
2
2
2
6 2
2
2 \
Now we can compute the local cohomology of normal semigroup rings. In the sequel ;C is the ane semigroup f;c : c 2 C g. Theorem 6.3.4. Let C be a positive normal semigroup of rank d, k a eld,
2
and z ZC. (a) If z relint( C ), then (L. )z is isomorphic to 0 k 0 with k in degree d. Consequently H i (L.)z = 0 for i = d, and H d (L. )z = k = (L. )z . (b) Suppose that z = relint( C ). Let T be a cross-section of R+C with face lattice , and = F T : F (R+C ) visible from z . Then
2
;
6
2 ; F S f \ 2F ; (i) (L )z = HomZ (Ce (F) Ce (S))(;1) k , (ii) He i (F) = He i (S) = 0 for all i,
! ! g
.
(iii) (H i (L ))z = 0 for all i. .
(a) For z 2 relint(;C ) one has z 2 RF if and only if F = R+C . (b)(i) The complex Ce (F) consists of direct summands Zf , f 2 F. As S is a cell subcomplex of F, Ce (S) is a chain subcomplex of Ce (F), and we obtain Ce (F)=Ce (S) if we replace all the direct summands Zf with f 2 S by 0. The complex HomZ ((Ce (F)=Ce (S))(;1) k) is therefore isomorphic to the complex Proof.
D. : 0
;! D0 ;!@ ;!@ Dd ! 0
272 with
6. Semigroup rings and invariant theory Dt =
M f2Ft;1 nS
kf and @((f 0) ) =
X
"(f f 0)f :
According to 6.3.3, (L )z is given by D . (ii) Note that the combinatorial structures of F and S do not depend on the chosen cross-section T . (This follows from 6.1.8.) Therefore we may vary T . Furthermore it was observed above that He i (F) = 0 for all i. If z 2 C , then S = , and Ce (S) is the zero complex. So suppose that z 2= C in the following. If z 2= ;C , then, by virtue of 6.1.13, there exists a hyperplane E through z which is transversal to R+C . Choose T = E \ R+C . Then S is the set of faces of T which are visible from z , and we invoke 6.3.1 in conjunction with 6.2.4 to conclude that He i (S) = 0 for all i. If z 2 ;C , then there exists a point z 0 2 RC n (;C ) with (z 0 ); = z ; . (By hypothesis z 2= relint(;C ) consider a suciently small neighbourhood of z .) Because of 6.3.2 we may replace z by z 0 in de ning S and argue as in the case z 2= ;C . (iii) We have a long exact sequence ;! He i(S) ;! He i(F) ;! He i(Ce (F)=Ce (S)) ;! He i+1(S) ;! Thus it follows from (ii) that Ce (F)=Ce (S) is exact. As it is a complex of free Z-modules, the dual (of a shifted copy) with respect to an arbitrary Z-module is also exact. The previous theorem allows us not only to show that normal semigroup rings are Cohen{Macaulay, but also to determine their canonical modules. Theorem 6.3.5. Let C be a normal semigroup, and k a eld. Then (a) (Hochster) kC ] is a Cohen{Macaulay ring, (b) (Danilov, Stanley) the ideal I generated by the monomials X c with c 2 relint C is the canonical module of kC ]. Proof. (a) We write kC ] in the form kC ] = kC0 ] kC 0 ] as in 6.1.3 0 ; 1 ; 1 then kC ] = kC ]X1 X1 . . . Xu Xu ] for some u 0. In view of 2.1.9 it is therefore enough to show that kC 0 ] is Cohen{Macaulay. But this follows immediately from 6.2.6 and 6.3.4, the latter of which in particular says that H i (L ) = 0 for i = 0 . . . d ; 1. (b) Suppose rst that C is positive. As Ld = kZC ], we have an exact sequence (1) 0 ;! U ;! kZC ] ;! H d (L ) ;! 0 of ZC -graded kC ]-modules. The functor Homk ( k) in the category of ZC -graded kC ]-modules assigns each module the k-vector space M Homk (M;z k) .
.
.
.
z2ZC
273
6.3. Normal semigroup rings
which is a ZC -graded kC ]-module in a natural way. Applying this functor to the exact sequence above we obtain an exact sequence 0 ;! Homk (H d (L ) k) ;! kZC ] ;! U 0 ;! 0: It follows from 6.3.4 that Homk (H d (L ) k) consists exactly of those graded components kZC ]z with z 2 relint C . Therefore I = Homk (H d (L ) k) as ZC -graded modules: As in the proof of 6.2.6 we use that kC ] has an admissible grading. Thus it is a local Z-graded ring whose maximal ideal is generated by the monomials X c, c 2 C , c 6= 0. Furthermore each Z-homogeneous component of kC ] is the direct sum of nitely many ZC -graded components. The same holds for H d (L ). As Homk commutes with nite direct sums, we conclude that .
.
.
m
.
as Z-graded modules: In Section 3.5 we de ned the local cohomology functors H i ( ) in the category of Z-graded kC ]-modules. If M is a Z-graded kC ]-module, then L M is a complex of Z-graded modules, and virtually the same arguments as in the proof of 6.2.5 show that H i (M ) = H i (L M ) for all i. Finally we deduce from 3.6.19 and 3.6.9 that I is the canonical module of kC ]. The general case of (b) in which C is not necessarily positive follows as in (a) if we use 3.3.21 to compute a canonical module of a polynomial extension. Corollary 6.3.6. Suppose, in addition to the hypothesis of 6.3.5, that C is positive. Then I is the (unique) canonical module of kC ] with respect to I = Homk (H d (L. ) k)
m
.
.
m
an arbitrary admissible grading.
Remark 6.3.7. The formulation `the canonical module' of 6.3.5 needs justi cation beyond 6.3.6. First, if we had developed the theory of Zn graded rings to the same extent as that of Z-graded rings, it would be immediate that I is the unique ZC -graded canonical module of kC ] (up to an isomorphism of ZC -graded modules). Second, and even more, a canonical module of kC ] is unique in the category of all kC ]-modules. We briey indicate the argument it exploits the theory of class groups (Fossum 108]), and will be explained in detail in Section 7.3 where it is more essential. With our usual notation, the extension kC 0 ] ! kC ] induces an isomorphism of class groups Cl(kC 0 ]) = Cl(kC ]). Because of this isomorphism a canonical module ! of kC ] is of the form !0 kC ] for some kC 0 ]-module !0 . The extension kC 0 ] ! kC ] is faithfully at. Applying 3.3.30 one concludes that !0 is a canonical module of kC 0 ]. Thus it is enough to consider positive semigroups C . For those one has
274
6. Semigroup rings and invariant theory
an isomorphism Cl(kC ]) = Cl(kC ] ) (108], 10.3). Finally one uses that the canonical module of a local ring is unique. The preceding argument amounts to the fact that a projective rank 1 module over kC ] is free. This was shown for arbitrary projective kC ]modules by Gubeladze 145]. Corollary 6.3.8. Let C be a normal semigroup, and k a eld. Then kC ] is Gorenstein if and only if there exists c 2 relint C with relint C = c + C. Proof. If relint C = c + C , then the ideal I of 6.3.5 is principal, and kC ] is Gorenstein by 3.3.7. For the converse implication we decompose C in the form C = C0 C 0 where C0 is a group and C 0 is positive. If kC ] is Gorenstein, then kC 0 ] is Gorenstein: the extension kC 0 ] ! kC ] is faithfully at, and the Gorenstein property descends from kC 0 ] to kC ] by 3.3.30. For kC 0 ] we can apply 3.6.11 (with respect to an admissible grading), and thus I = kC 0 ]. It follows that I is generated by an element c X . Therefore relint C 0 = c+C 0 , and it is easy to verify that relint C = c+C as well. m
S S
Combinatorial applications. Let be a system of homogeneous linear Diophantine equations in n variables. It follows directly from 6.1.2 that the set C of solutions c Nn of is a positive normal semigroup. This
2
fact enables us to apply results on Hilbert functions to the combinatorial object C . The set C can be represented by the power series C (t ) =
X c2C
tc
in n variables t = t1 . . . tn. Obviously C (t ) is the Zn-graded Hilbert series of kC ] if we consider the Zn-grading on kC ] it inherits from kNn ] = kX1 . . . Xn]. As we have not developed the theory of Zn -graded modules to the necessary extent, we restrict ourselves to considering the specialization X c(t) = tjcj: c2C
It is the Hilbert series of kC ] for the Z-grading induced by the total degree of a monomial. Under this grading kC ] is a positively graded k-algebra. Let C + be the set of strictly positive integral solutions of S, i.e. solutions c 2 Nn with ci > 0 for i = 1 . . . n. It may of course happen that C + = , but otherwise we have C + = relint C (Exercise 6.3.14). Therefore, and by 6.3.6, the power series c+(t) =
X
c2C +
tjcj
275
6.3. Normal semigroup rings
is the Hilbert series of the canonical module ! of kC ]. Hence 4.4.6 immediately yields the following reciprocity law . Theorem 6.3.9 (Stanley). With the notation introduced, suppose that C + is non-empty. Then
;
c+(t) = ( 1)d c(t;1)
d = rank C:
Of all the results of Section 4.4 only 4.4.6 has been applied to kC ]. We could extend 6.3.9 by furthermore considering the Hilbert function H (kC ] m) = jfc 2 C : jcj = mgj. Below, such an extension is carried out for the Ehrhart function of a rational polytope. Remark 6.3.10. In 361], Theorem 4.6.14, Stanley proves the ` ne' version C +(t ) = (;1)d C (t ;1) of the previous theorem by combinatorial methods. In order to obtain it by ring-theoretic arguments one needs the Zn -graded variant of 4.4.6 which was also given by Stanley see 357], Theorem 6.1. (Exercise 5.6.6 is the Zn -graded variant of 4.4.6 for Stanley{Reisner rings.) Conversely, the computation of the canonical module of a normal semigroup ring in 357] uses the ne reciprocity law: similarly as in 5.6.3 one shows that the ideal generated by the monomials X c, c 2 C +, is the canonical module of kC ] once the equation C +(t ) = (;1)d C (t ;1) has been established. Let P Rn be a polytope of dimension d . Since P is bounded, we may de ne its Ehrhart function by
jf 2 Zn : mz 2 P gj
E (P m) = z
and its Ehrhart series by
EP (t) =
2 N m > 0
m
X m2N
and E (P 0) = 1:
E (P m)tm:
It is clear that E (P m) = jfz 2 Zn : z 2 mP gj where mP = fmp : p 2 P g. Similarly as above we set
jf 2 Zn : mz 2 relint P gj
E +(P m) = z
and
EP+(t) =
X
m2N
for m > 0, E +(P 0) = 0
E +(P m)tm:
Note that E +(P m) = jfz 2 Zn : z 2 relint mP gj for m > 0. We de ne the cone D Rn+1 by D = R+f(p 1): p 2 P g: Then C = D \ Zn+1 is a subsemigroup of Zn+1. Therefore one may consider the
276
6. Semigroup rings and invariant theory
k-algebra kC ]. Suppose P is a rational polytope then D is a rational cone, and C is a positive normal semigroup. Let us x a grading on kC ] by assigning to c = (c1 . . . cd +1) the degree cd +1. For this grading the Hilbert functions of kC ] and of the ideal I generated by the monomials X c, c relint C , are given by
2
H (kC ] m) = E (P m) and H (I m) = E +(P m):
The grading under consideration is admissible for kC ], and therefore we may apply the theory of Chapter 4 to kC ]. Part (b) of the following theorem is Ehrhart's remarkable reciprocity law for rational polytopes. Theorem 6.3.11 (Ehrhart). Let P Rn be a d-dimensional rational polytope, d > 0. Then (a) EP (t) is a rational function, and there exists a quasi-polynomial q with E (P m) = q(m) for all m 0 (b) EP+(t) = (;1)d +1EP (t;1), equivalently
;
;
E +(P m) = ( 1)d E (P m)
;
;
for all m
1
where E (P m) = q( m) is the natural extension of E (P ). Proof. (a) Since EP (t) is the Hilbert series of a positively graded Noetherian k-algebra, it is a rational function. According to 4.4.3 we must show for the second statement in (a) that EP (t) has negative degree, or, equivalently, that the a-invariant of kC ] is negative. By 6.3.5 the ring kC ] is Cohen{Macaulay, and by 6.3.6 its canonical module is generated by the elements X c, c relint C . These have positive degrees under the grading of kC ], and hence a(kC ]) < 0. (b) By what has just been said, EP+(t) is the Hilbert series of the canonical module of kC ]. Furthermore, dim kC ] = d + 1. Thus the
2
rst is a special case of 4.4.6. The second equation results from P equation E ( P ; m )tm = ;EP (t;1). The reader may prove this identity as an m 1 exercise, or look up 361], 4.2.3. The quasi-polynomial q in 6.3.11 is called the Ehrhart quasi-polynomial of P . Suppose that P is even an integral polytope, that is, a polytope whose vertex set V is contained in Zn. Then, in addition to kC ], we may also consider its subalgebra kV ] = kX (v1) : v 2 V ]: Obviously kV ] is a homogeneous k-algebra. Let c 2 C then there exist P qv 2 Q+ such that c = v2V qv v. If we multiply this equation by a suitable common denominator e and interpret the result in terms of monomials, then we see that (X c)e 2 kV ]. Thus kC ] is integral over kV ]. Since it is
277
6.3. Normal semigroup rings
also a nitely generated kV ]-algebra, it is even a nite kV ]-module. In particular, by Hilbert's theorem 4.1.3, the Ehrhart quasi-polynomial of P is a polynomial and therefore called the Ehrhart polynomial. Furthermore kC ] has a well de ned multiplicity. In concluding this section we want to illuminate the beautiful relation between the volume vol P of an ndimensional integral polytope P Rn and the multiplicity of kC ]. Theorem 6.3.12. Let P Rn be an n-dimensional integral polytope, and let kC ] the normal semigroup ring constructed above. Then e(kC ]) = n! vol P : Proof.
of P is
Elementary arguments of measure theory show that the volume vol P = lim E (P m) : m!1
mn
Being the Hilbert polynomial of a (n + 1)-dimensional kV ]-module, E (P m) has degree n. Thus its leading coecient is given by vol P . On the other hand, it is also given by e(kC ])=n!. The restriction to n-dimensional polytopes P Rn is only for sim-
plicity see 361], Section 4.6, for the general case. Using the fact that the volume of P is the leading coecient of its Ehrhart polynomial one can derive classical formulas for vol P . Exercise 6.3.17 presents the cases n = 2 and n = 3. Exercises
6.3.13. Let C be a positive normal semigroup. For each i = 0 . . . d let Gi be the direct sum of the residue class rings k C ]= (F ) where F is an i-dimensional face of R+C . De ne the map k C ]= (F ) ! k C ]= (F ) to be "(f f ) nat if F F , dim F = dim F ; 1, or 0 otherwise. Show that the induced sequence P
0
P
0
0
P
0
0 ;;! I ;;! k C ] = Gd ;;! Gd 1 ;;! ;;! G1 ;;! G0 = k ;;! 0 is exact (of course I is de ned as in 6.3.5). Hint: The proof is similar to that of 5.7.3. 6.3.14. Let C be the semigroup of solutions c 2 Nn of a system of homogeneous linear Diophantine equations, and C + = fc 2 C : ci > 0 for all ig. Show that if C + 6= , then C + = relint C . 6.3.15. Let P be an integral polytope of dimension n, and de ne the semigroup C and the grading of k C ] as above. It is customary to call (h0 . . . hn ) the h-vector of P where hi is the i-th coe cient of the (Laurent) polynomial Q(t) in the numerator of the Ehrhart series of k C ] it follows from 6.3.11 that hi = 0 for i > n. Prove the following inequalities due to Stanley 362] and Hibi 168]: (a) hPi 0 for P all i (b) ji=0 hi ji=0 hs i for all j = 0 . . . s where s = maxfi : hi 6= 0g nat
;
;
278
6. Semigroup rings and invariant theory
+1 (c) ni=n j hi ji=0 hi for all j = 0 . . . n. Hint: For (b) and (c) study (again) the proof of 4.4.9. For R = k C ] we also have an exact sequence 0 ! ! ! R ! R=! ! 0. 6.3.16. With P , C , and k C ] as in 6.3.15, set L = f(p 1): p 2 P \ Zn g and let k P ] be the k-algebra generated by the elements X w , w 2 L. (a) Show the following are equivalent: (i) k C ] = k P ] (ii) k C ] is homogeneous (iii) k P ] is normal and ZL = Zn+1 . (b) Discuss the conditions of (a)(iii) for the polytopes P1 P2 R3 spanned by (1) v0 = 0, v1 = (1 0 0), v2 = (0 1 0), and v3 = (1 1 3), and (2) v0 = 0, v1 = (2 0 0), v2 = (0 3 0), and v3 = (0 0 5). 6.3.17. Prove that the volume of an n-dimensional integral polytope P in Rn is
P
P
;
vol P = 12 (E (P 1) + E +(P 1) ; 2) for n = 2, and vol P = 61 (E (P 2) ; 3E (P 1) ; E +(P 1) + 3) for n = 3. Hint: The coe cients of a polynomial can be determined by interpolation.
6.4 Invariants of tori and nite groups In the following we use some elementary notions and results from the theory of linear algebraic groups for which we refer the reader to Humphreys 208], Kraft 241], or Mumford and Fogarty 279]. Let k be an algebraically closed eld, and V a k-vector space of nite dimension. Each ' 2 GL(V ) yields a k-algebra automorphism ' of the symmetric algebra R = S (V ). In concrete terms, if e1 . . . en is a basis of V , then S (V ) = kX1 . . . Xn], the isomorphism being induced by the linear map which sends ei to Xi , i = 1 . . . n. If we identify V and kX1 + + kXn via this map, then ' is just the k-algebra automorphism of R = kX1 . . . Xn] given by the substitution Xi 7! '(Xi). (From a categorical point of view it would be better to consider the action of GL(V ) on S (V ), the ring of polynomial functions on V .) Suppose that G is a linear algebraic group over k such a group is always isomorphic to a Zariski closed subgroup of GL(W ) where W is a suitable nite dimensional k-vector space. A morphism : G ! GL(V ) (in the category of algebraic groups) is called a representation of G. It assigns the automorphism (g) of R to each g 2 G, so that we say that G acts linearly on R . It is the classical problem of (algebraic) invariant theory to determine the structure of the ring of invariants
f 2 R:
RG = f
g(f ) = f for all g
2 Gg
279
6.4. Invariants of tori and nite groups
where we have set g(f ) = (g)(f ) for simplicity of notation. If f is homogeneous of total degree d , then so is g(f ). Therefore R G is a positively graded k-algebra inheriting its grading from R . A character of G is a representation : G ! GL(k). To each character we associate the set R = ff 2 R : g(f ) = (g)f for all g 2 Gg of semi-invariants of weight . It is easily veri ed that R is a graded R G-submodule of R . Especially important in the following is the inverse determinant character g 7! det;1(g) = det (g);1 associated with . The ring R G of invariants only depends on (G) GL(V ) thus we shall often simplify the situation by directly considering a subgroup of GL(V ). Furthermore, for concrete groups the requirement that k be algebraically closed can sometimes be relaxed. More generally, one may always form the ring R G when R is a ring and G is a subgroup of Aut R . Clearly R G inherits all properties of R which descend to subrings, and is a normal domain along with R : Proposition 6.4.1. Let R be a normal domain, and G a subgroup of Aut R. Then R G is a normal domain. Proof. It is easy to see that R G is the intersection of its eld of fractions Q(R G) with R (within Q(R )).
Invariants of diagonalizable groups. Let k be an algebraically closed eld. For each m N the group GL(k)m is called a torus it is isomorphic to the group of m m diagonal matrices of rank m over k. Slightly more generally we want to consider diagonalizable groups over k, i.e. direct
2
products
D=T
H
where T is a torus and H is a nite Abelian group whose order is not divisible by char k. Since k contains a primitive q-th root of unity for each q not divisible by char k, H may be written in the form H = h'1i h'w i where h'j i is the cyclic subgroup of GL(k) generated by a root of unity 'j . Thus we may write each element in D in the form (d1 . . . dm '1s . . . 'wsw ) with sj 2 N. Suppose now that we are given a representation of D, that is, a homomorphism : D ! GL(V ). Then can be diagonalized: there exists a basis e1 . . . en of V such that each ei is an eigenvector of (d ) for every d 2 D. Thus the vector subspace kei = k is stable under the action of D, and therefore induces a character i of D i associates to each element d 2 D its eigenvalue with respect to ei . It is sucient to 1
280
6. Semigroup rings and invariant theory
determine the characters of the direct factors GL(k) and h'j i of D. One sees easily that in both cases the characters are the powers a 7! as , s 2 Z. Thus there exist t1i . . . tmi 2 Z and u1i . . . uwi 2 N, i = 1 . . . n, such that (d1 . . . dm '1s . . . 'wsw )(ei) = d1t i dmtmi '1s u i 'wsw uwi ei for i = 1 . . . n. Theorem 6.4.2. Let k be an algebraically closed eld, and D a diagonalizable group over k acting linearly on a polynomial ring R = kX1 . . . Xn]. 1
1
1 1
Then
(a) (Hochster) the ring R D of invariants is a graded Cohen{Macaulay ring, (b) (Danilov, Stanley) R det; (;n) is the canonical module of R D , provided ; R det 6= 0. Proof. We may assume right away that D acts diagonally as just described. It follows that each monomial X1a Xnan is mapped to a multiple of itself by every d 2 D. Therefore f 2 R is invariant if and only if all its monomials are invariant, so that R D = kC ] for some semigroup C Nn. Extending the formula for the action of D to monomials, we see that X1a Xnan is an invariant if and only if (a1 . . . an) satis es the system tj 1a1 + + tjnan = 0 j = 1 . . . m of homogeneous linear equations with integral coecients, and simultaneously the system sj (uj 1a1 + + ujnan) 0 mod (ord 'j ) j = 1 . . . w of homogeneous congruences (of course ord 'j denotes the order of the root of unity 'j ). It follows easily that C is the intersection of Rn+ with a nitely generated group G Qn. Therefore C is normal, and part (a) is an immediate consequence of 6.3.5(a). Similarly, part (b) can be derived rather quickly from 6.3.5(b). Set ; S = R D , M = R det , and P = X1 Xn. Then d (P ) = det(d )P for all d 2 D. Hence, for every f 2 R , f 2 M if and only if P f 2 S . Obviously M is an S -module generated by monomials (even as a k-vector space), and therefore a graded S -module. Let I be the ideal generated by the monomials X1c Xncn with (c1 . . . cn ) 2 relint C . We know from 6.3.5 in conjunction with 6.3.7 that I is the graded canonical module of S (up to an isomorphism of graded modules). Evidently it is enough to show that P M I and P ;1I M , provided M 6= 0. The representation C = Rn+ \ G readily yields that f(c1 . . . cn) 2 C : ci > 0 for all ig relint C: 1
1
1
1
1
1
281
6.4. Invariants of tori and nite groups
Therefore P M I . Conversely, suppose M 6= 0, and let X1b Xnbn 2 M . Then P X1b Xnbn 2 S , and so C contains an element (c1 . . . cn ) with ci > 0 for all i. Hence C is not contained in a coordinate hyperplane, and consequently no relative interior point of C lies in such a hyperplane. It follows that P ;1I kX1 . . . Xn], and thus P ;1I M . The preceding proof shows that the degenerate case R det; = 0 occurs precisely when, after diagonalization, R G is contained in one of the subrings kX1 . . . X^ i . . . Xn]. Furthermore, the condition that k be algebraically closed is dispensable once the action of D is a priori diagonal. If k is in nite, then the proof of 6.4.2 remains valid without modi cation if k is nite, then one must set T = fidg and m = 0. This generalization can also be extended to the following corollary. Corollary 6.4.3. Under the hypothesis of 6.4.2 suppose additionally that det d = 1 for all d 2 D. Then R D is a Gorenstein ring. The proof of 6.4.2 suggests that 6.4.2 is just a special case of 6.3.5 however, Exercise 6.4.16 shows that these theorems are actually equivalent. 1
1
1
Finite groups. Theorem 6.4.2 in particular covers the case in which a nite Abelian group G acts linearly on a polynomial ring kX1 . . . Xn], provided the order G of G is invertible in k. With the same proviso, we
jj
now want to treat the case of an arbitrary nite group. It is convenient to restrict oneself to subgroups G of GL(V ). More generally let us rst consider a ring R and a nite group G of automorphisms of R such that jGj is invertible in R . Let S be the ring R G of invariants, and set
j j X g(r)
(r) = G ;1
g2G
for every r 2 R . It is straightforward to verify that is an S -linear map from R to S with jS = idS . A map satisfying these conditions is called a Reynolds operator (for the pair (R S ).) The existence of a Reynolds operator is obviously equivalent to the fact that S is a direct summand of R as an S -module. Proposition 6.4.4. Let R be a ring, S a subring of R, and suppose that there exists a Reynolds operator for (R S ). Then the following hold: (a) for every ideal I of S one has IR \ S = I (b) if R is Noetherian, then so is S (c) if x is an R-sequence in S, then it is also an S-sequence. P Proof. (a) P For s1 . . . sn 2 S , r1 . . . rn 2 R with r = si ri 2 S one has r = (r) = si (ri).
282
6. Semigroup rings and invariant theory
(b) If I0 I1 is an ascending sequence of ideals in S , then the sequence I0 R I1 R is stationary in R . Therefore, and by (a), the sequence I0 I1 is also stationary. (c) This follows easily from (a). In the case of a group action considered above, each r 2 R is a solution of the equation
Y
(X ; g(r)) = 0:
g2G
The left hand side is a monic polynomial in X whose coecients are elementary symmetric functions in the elements g(r), g 2 G. Therefore all the coecients belong to the ring S of invariants, and we see that R is integral over S . Theorem 6.4.5 (Hochster{Eagon). Suppose R is a Cohen{Macaulay ring and S is a subring such that there exists a Reynolds operator , and R is integral over S. Then, if R is Cohen{Macaulay, so is S. Proof. We must show that the localizations S of S with respect to its maximal ideals are Cohen{Macaulay. Given a maximal ideal of S , we replace S by S and R by R S . Therefore we may assume that S is a local ring with maximal ideal . Since R is integral over S , it is a n
n
n
n
n
semi-local ring. (This follows easily from A.6.) We argue by induction on the length of a maximal R -sequence in . Suppose rst that consists entirely of zero-divisors of R . Then each s 2 is contained one of the associated prime ideals 1 . . . m of S \ Sin, and R . So = there exists a j with = j \ S . As R is i Cohen{Macaulay, all the i are minimal prime ideals of R . On the other hand, since R is integral over S , j is also a maximal ideal of R . If j is the only maximal ideal of R , then it follows immediately that dim S = dim R = 0 so that S is Cohen{Macaulay as desired. Otherwise the zero ideal of R can be written \ where is j -primary and 6 j . As + = R , the Chinese remainder theorem implies that R splits into the direct product of subrings R1 , R2. If we can replace R by one of them, then we can nish the case under consideration by induction on the number of maximal ideals of R . Let 1 and 2 be the projections of R onto R1 and R2 , and the embedding of S into R . Then both 1 and 2 are endomorphisms of the S -module S . Hence there exist s1 and s2 such that i is multiplication by si . It follows that 1 = (1) = ( 1 + 2) (1) = s1 + s2 so that at least one of s1 and s2 is a unit in S , say s1 . Then 1 is an embedding of S into R1, and one easily checks that all the hypotheses n
n
n
n
p
n
p
n
p
p
p
p
p
q
q
r
r
q
p
r
p
283
6.4. Invariants of tori and nite groups
pass on to the pair (R1 S ). This nishes the case in which consists entirely of zero-divisors of R . Now suppose that s 2 is R -regular. Then 6.4.4 implies that S=sS is in a natural way a subring of R=sR , and it is again easily veri ed that the remaining hypotheses hold for the subring S=sS of R=sR . As R=sR is Cohen{Macaulay, we conclude that S=sS and, hence, S are Cohen{ Macaulay. Corollary 6.4.6. Let R be a Cohen{Macaulay ring, and G a nite group n
n
of automorphisms of R whose order is invertible in R. Then the ring R G of invariants is Cohen{Macaulay. Remark 6.4.7. In our derivation of 6.4.6 we have used that G is invertible in R in order to show that R G is Noetherian. However, for this property of R G the hypothesis on G is quite inessential: if R is a nitely generated algebra over a Noetherian ring k such that G acts trivially on k, then, by a famous theorem of E. Noether, R G is a nitely generated k-algebra. We saw above that R is integral over R G . Therefore R is already integral over the k-subalgebra A generated by the coecients of the equations fi(x) = 0, fi kT ], which establish that the nitely many generators xi of R are integral over R G . It follows that R and, hence, R G are nite A-modules. On the other hand, that G is invertible in R is essential for the Cohen-Macaulay property of R G . In fact, if k is a eld of characteristic 2, then kX1 . . . X4]G is a non{Cohen{Macaulay factorial domain for the group G of cyclic permutations of X1 . . . X4 see Bertin 41]. Similarly to 6.4.2 one can determine the canonical module of R G from invariant theoretic data if G acts linearly on a polynomial ring R . Let V be again a vector space of nite dimension over a eld k which we now
jj
jj
2
jj
assume to be of characteristic 0 (see Remark 6.4.11 for the more general case in which jGj is not divisible by char k). As above we extend the action of G to the symmetric algebra R = S (V ) which we may identify with the polynomial ring kX1 . . . Xn], n = dim V , whenever it is appropriate. Let S = R G . Since the action of G can be restricted to the graded components Ri of R , S is a positively graded k-algebra. Being a nitely generated integral extension of S , R is a nite graded S -module, and in fact a maximal Cohen{Macaulay S -module: according to 1.5.17 there exists a homogeneous system of parameters x in S it follows that height xR = n, and thus x is an R -sequence. (In conjunction with 6.4.4 this observation yields a quick proof of the previous corollary in the special case under consideration.) It is customary to call the Hilbert series of S the Molien series of G: MG (t) = HS (t) =
1 X i=0
dim Si ti:
284
6. Semigroup rings and invariant theory
We also need the Molien series and the Reynolds operator for the semiinvariants of G. For a character of G we set M (t) = HM (t) =
1 X i=0
dim Riti and (r) = jGj;1
X
g 2G
(g);1g(r):
It is easy to check that (R ) = R and (r) = r for r 2 R . The operator is a k-endomorphism of the graded k-vector space R . Let i denote its restriction to Ri then i = (i)2 , and therefore dim Ri = dim Im i = Tr i = jGj;1
X
g2G
j
(g);1 Tr g Ri :
Here Tr denotes the trace, and we use its linearity. Combining the formulas yields 1 j j X (g);1 X(Tr gjR )ti:
M (t) = G ;1
g 2G
i
i=0
Theorem 6.4.8 (Molien's formula). Let k be a eld of characteristic 0, V a nite dimensional k-vector space, and G a nite subgroup of GL(V ). Then the Molien series of a character of G is given by (g);1 j j X det(id ;tg) :
M (t) = G ;1 Proof.
g2G
We need to show that
1 X 1 = (Tr gjRi )ti det(id ;tg) i=0
for each g 2 G. In fact, this equation holds for an arbitrary element g 2 GL(V ). In order to prove it we may extend k to an algebraically closed eld. Then, for a suitable basis X1 . . . Xn of V , g is given by an upper triangular matrix whose diagonal entries are the eigenvalues 1 . . . n of g (as an element of GL(V )). The monomials of total degree i in X1 . . . Xn form a basis of the vector space Ri . If these monomials are ordered lexicographically, then gjRi is again represented by an upper triangular matrix whose diagonal entry corresponding to the monomial X a = X1a Xnan is a = a1 ann . Therefore X Tr gjRi = a 1
jaj=i
1
6.4. Invariants of tori and nite groups
285
and the expansion of the product of the geometric series 1=(1 ; j t), j = 1 . . . n, gives us 1 1 X X X Yn 1 (Tr gjRi )ti = ati = 1 ; j t : i=0 j =1 i=0 jaj=i Using that ;1 1 . . . ;n 1 are the eigenvalues of g;1 , we nally get Yn 1 Yn ;j 1 1 det g;1 = = = : ; 1 ; 1 1 ; j t j =1 j ; t det(g ; t id) det(id ;gt) j =1
We can now easily prove the analogues of 6.4.2 and 6.4.3 for linear actions of nite groups: Theorem 6.4.9 (Watanabe). Let k be a eld of characteristic 0, V a kvector space of dimension n, R = S (V ), and G a nite subgroup of GL(V ). (a) Then R det; (;n) is the canonical module of R G . (b) In particular R G is Gorenstein if G SL(V ). Proof. Set S = R G and = det;1. Since is an S -linear map from R onto N = R , we see that N is a direct S -summand of R . It was observed above that R is a maximal Cohen{Macaulay module over S therefore N is also a maximal Cohen{Macaulay S -module. Furthermore X det g X 1 = j Gj;1 M (t) = jGj;1 ; 1 det(id ;tg) det(g ; t id) g 2G g 2G X (;1)nt;n X 1 = jGj;1 = jGj;1 det(g ; t id) det(id ;t;1 g) g 2G g 2G 1
= (;1)nt;nMG (t;1): As the Molien series are Hilbert series, we may apply 4.4.5 to conclude that N (;n) is the canonical module of S . This proves (a). If G SL(V ), then, by (a), S is isomorphic to the canonical module of S . As a canonical module is canonical, S is Gorenstein. Very easy examples show that 6.4.9(b) cannot be reversed. The obstruction is the presence of pseudo-reexions in G: g 2 GL(V ) is called a pseudo-reexion if it has nite order and its eigenspace for the eigenvalue 1 has dimension dim V ; 1. (Thus the remaining eigenvalue is the determinant.) Theorem 6.4.10. With the notation of 6.4.9 the following hold. (a) (Stanley) R G is Gorenstein if and only if X 1 X det g = t;m det(id ;tg) det(id ;tg) g2G g2G where m is the number of pseudo-reexions in G.
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6. Semigroup rings and invariant theory
(b) (Watanabe) Suppose G contains no pseudo-reexions. Then R G is Gorenstein if and only if G SL(V ). Proof. (a) If we apply 4.4.6 to the Molien series of R G, then it follows easily that R G is Gorenstein if and only if the equation in (a) holds for some m 2 Z. It remains to determine m. To this end we expand both sides in a Laurent series at t = 1. Let n = dim V , and denote the set of pseudo-reexions in G. The pole order of 1= det(id ;tg) at t = 1 is the multiplicity of 1 as an eigenvalue of g. Thus the only summand with a pole of order n is 1= det(id ;t id) = 1=(1 ; t)n , and those with a pole of order n ; 1 are exactly the summands 1 1 1 + = 2 det(id ;t) (1 ; t)n;1 1 ; det where denotes terms of higher order in (1 ; t). Thus the left hand side is X 1 1 1 + (1 ; t)n (1 ; t)n;1 2 1 ; det + whereas the right hand side is
X (1 + m(1 ; t) + ) (1 ;1 t)n + (1 ;1t)n;1 1 ;detdet + 2
so that a comparison of coecients yields m = j j as required.P (b) Evaluating the formula in (a) for t = 0 gives jGj = g2G det g. Since the eigenvalues of the elements of G are roots of unity, we must have det g = 1 for all g 2 G. (Note that the elements of k which are algebraic over k may be considered complex numbers.) Remark 6.4.11. Theorems 6.4.9 and 6.4.10 were proved by Watanabe 387], 388] under the weaker assumption that jGj is not divisible by char k. His proofs use divisorial methods. Hini%c 172] extended Watanabe's results to invariant subrings of Gorenstein rings. Finite groups generated by pseudo-reexions. That the pseudo-reexions in a nite group G GL(V ) play a special role has already been
demonstrated by 6.4.10. However, the most ostensive indication of this fact is the celebrated theorem which characterizes the regular ones among the rings of invariants of nite groups: Theorem 6.4.12 (Shephard{Todd, Chevalley, Serre). Let k be a eld of characteristic 0, V a k-vector space of dimension n, R = S (V ), and G a nite subgroup of GL(V ). Then the following are equivalent: (a) G is generated by pseudo-reexions
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6.4. Invariants of tori and nite groups
(b) R is a free R G-module (c) the k-algebra R G is generated by (necessarily n) algebraically independent elements.
That (c) is equivalent with the regularity of R G follows from Exercise 2.2.25, which also shows that the algebraically independent elements can be chosen homogeneous their number is n, as dim R G = dim R = dim V = n. The remainder of this section is devoted to a proof of 6.4.12. The next lemma covers the equivalence (b) () (c). Lemma 6.4.13. Let R be a positively graded, nitely generated algebra
over an arbitrary eld k, and S a graded k-subalgebra such that R is a nite S-module. (a) Then S is a nitely generated k-algebra. (b) If R is Cohen{Macaulay and S is generated by algebraically independent elements over k, then R is a free S-module. Moreover, it has a basis of homogeneous elements. (c) If R is generated by algebraically independent elements over k and a free S-module, then S is generated by algebraically independent elements.
(a) This is a special case of E. Noether's theorem proved in 6.4.7. (b) By hypothesis S is a regular ring a minimal homogeneous system x1 . . . xn of generators of its maximal ideal is algebraically independent, and furthermore generates S as a k-algebra (see Exercise 2.2.25). Since R is a nite S -module, x1 . . . xn is also a homogeneous system of parameters of R , and thus an R -sequence by hypothesis on R . Consequently R is a maximal Cohen{Macaulay S -module. It follows from 2.2.11 that R is a projective S -module, and then 1.5.15 implies that R is a free S -module, and that every minimal homogeneous system of generators of R over S is a basis. (c) Let and be the maximal ideals of R and S . The hypothesis implies that R is a regular local ring and a at local extension of S . Thus S is regular according to 2.2.12. Again we apply 2.2.25 to conclude that S is generated by algebraically independent elements. We now show that (a) ) (b) in 6.4.12. For R to be a free S -module, S = R G , it is sucient that M = TorS1 (R S= ) = 0. In fact this implies that R is a free S -module, whence R is free over S by 1.5.15. The module M is the kernel of the homomorphism ' : R S ! R S S induced by the embedding S . Given a minimal homogeneous x1 . . . xm P yi system of generators of , M consists of all the elements x , y 2 R with i i P yixi = 0. Evidently M is a graded submodule of the graded S -module R with deg a b = deg a +deg b for homogeneous elements a 2 R , b 2 . We assume that M 6= 0. In order to derive a contradiction choose a non-zero Proof.
m
n
m
n
n
n
n
n
n
n
n
n
n
288
6. Semigroup rings and invariant theory
P
homogeneous element yi xi of minimal degree in M . Replacing yi by a suitable homogeneous component, we may suppose that each yi is itself homogeneous.P P We claim that yi xi = g(yi) xi for all g 2 G. Since G is generated by pseudo-reexions, it is enough to show this for a pseudoreexion . We choose a basis X1 . . . Xn of V such that (Xi ) = Xi for i = 2 . . . n and (X1) = 'X1 where ' is a root of unity. For each monomial f in X1 . . . Xn it follows easily that X1 divides (f ) ; f . Therefore X1 divides (f ) ; f P for every element P f of R . Let (yi ) ; yi = X1yi0 for 0 i =P 1 . . . n. Then yi xi =P0 so that yi0 xi 2 MP. From the P assumption on yi xi we conclude yi0 xi = 0 and, hence, yi xi = (yi ) xi . The Reynolds operator , viewed as an S -endomorphism of R , induces an S -linearPmap 0 = P id: R ! R . By what has just been proved, 0 ( yi xi ) = yi xi . On the other hand Im = S so that 0 factors as n
n
;! S ;! R where is induced by the embedding S ! R . It is immediate that 00(M ) is mapped to the kernel TorS1 (S S= ) = 0 of the Pnatural map P 0 S ! S S = S . Thus ( yi xi ) = 0, and therefore yi xi = 0, which is the required contradiction. It remains to prove the implication (c) ) (a) for which we use a
R
00
n
n
n
n
n
combinatorial argument based on the following lemma.
Lemma 6.4.14. Let k be a eld of characteristic 0, V a k-vector space of nite dimension, R = S (V ), and G a nite subgroup of GL(V ). Let x1 . . . xn , n = dim V , be a homogeneous system of parameters of R G . (a) Then x1 . . . xn are algebraically independent over k, and R G is a free kx1 . . . xn]-module it has a basis of homogeneous elements h1 . . . hm. (b) Let di = deg xi , i = 1 . . . n, and ej = deg hj , j = 1 . . . m, and let denote the set of pseudo-reexions in G. Then
jj
m G = d1
dn
jj
and m + 2(e1 +
+ em) = m(d1 + + dn ; n):
Proof. (a) According to 1.5.17 R G is a nite kx1 . . . xn ]-module. Thus we have dim kx1 . . . xn ] = n so that x1 . . . xn are algebraically independent
over k. One now applies 6.4.13. Q (b) The Hilbert series of kx1 . . . xn ] is 1=L ni=1(1 ; tdi ). Thus the Hilbert series of the kx1 . . . xn]-module R G = hi kx1 . . . xn] is MG (t) =
e Qn (1;+tdt ) = (1 ;1 t)n f(t)
te +
m
1
i=1
i
6.4. Invariants of tori and nite groups
Q P
289
i ;1 j where f (t) = (te + + tem )= ni=1 dj =0 t does not have a pole at t = 1. Expansion in a Laurent series at t = 1 yields 1 ;f (1) ; f 0(1)(1 ; t) + MG (t) = (1 ; t)n with f (1) = m=(d1 dn ) and P e1 + + em ; (m=2) ni=1(di ; 1) 0 f (1) = : 1
d1
dn
As we saw in the proof of 6.4.10, we also have
1 X 1 1 + + : MG(t) = jGj;1 (1 ; t)n (1 ; t)n;1 2 1 ; det Observe that X 1 X 1 X 1 1X 1 =1 + = 1 = j j: 1 ; det 2 2 1 ; det 2 1 ; det ;1 2 2 2 2 Comparing coecients in the Laurent expansions gives the required formulas. We now complete the proof of 6.4.12 with the implication (c) ) (a). Let H be the subgroup of G generated by the pseudo-reexions in G. Using the implication (a) ) (c), we see that R H is generated by algebraically independent homogeneous elements y1 . . . yn. Since R G is, by hypothesis, also generated by algebraically independent homogeneous elements x1 . . . xn , we have an inclusion kx1 . . . xn] ky1 . . . yn]: We want to show that there exists a permutation of f1 . . . ng such that deg xi deg y(i) for all i. To this end we de ne Pi to be the smallest subset of f1S . . . ng such that xi 2 kyj : j 2 Pi ]. For each subset I of f1 . . . ng the set i2I Pi must have at least jI j elements since the xi , i 2 I , are algebraically independent. Thus the marriage theorem of elementary combinatorics guarantees an injective map with (i) 2 Pi for all i. By de nition of Pi we have deg xi deg y(i). Arranging y1 . . . yn in the order prescribed by we may assume that di = deg xi zi = deg yi for all i. Lemma 6.4.14 applied to R G = kx1 . . . xn] yields j j = d1 + + dn ; n since we have m = 1, h1 = 1, and d1 = 0. But 6.4.14 also applies to R H = ky1 . . . yn ], and since H contains all the pseudo-reexions of G, we similarly obtain j j = z1 + + zn ; n: Summing up, we must have di = zi for all i. Therefore jGj = jH j by the rst equation in 6.4.14, and G = H .
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Remarks 6.4.15. (a) The equivalence of (b) and (c) in 6.4.12 is independent of any assumption on the characteristic of k or the order of G, as is clearly exhibited by 6.4.13. Furthermore, the proof of the implication (a) ) (a) only uses that jGj is not divisible by char k. It is due to Serre 335], as well as a proof of (c) ) (a) (based on rami cation theory) which does not require any assumption on k or G see also Bourbaki 49], Ch. 5. For (c) ) (a) we have reproduced the original argument of Shephard and Todd 345] which exploits the fact that char k = 0 in an essential way. (b) Within the hierarchy `regular, complete intersection, Gorenstein, Cohen{Macaulay', the property of being a complete intersection is the most dicult for rings of invariants of linear actions of nite groups. A necessary condition for R G to be a complete intersection was given by Kac and Watanabe 228]: if R G is a complete intersection, then G is generated by elements g with rank(g ; id) 2. The proof uses geometric methods. Exercise 6.4.21 presents an example showing that this condition is not sucient for R G to be a complete intersection. See Gordeev 129] and Nakajima and Watanabe 287] for a classi cation of the groups G for which R G is a complete intersection. Nakajima 286] has classi ed the hypersurface rings R G . Exercises
Let S , T be a ne semigroups, S T . One says that S is an expanded subsemigroup of T if S = T \ QS (in QT ). Prove: (a) An expanded subsemigroup is a full subsemigroup. (b) The following are equivalent for a subsemigroup S of Nn : (i) S is expanded (ii) there exists a vector subspace U of Qn with S = U \ Qn (iii) there exists a homogeneous system of linear equations with integral coe cients such that S is the set of its non-negative solutions (iv) k S ] is the ring of invariants of a linear torus action on k X1 . . . Xn ]. (c) Every positive normal semigroup C is isomorphic to an expanded subsemigroup of Nn for some n 0. Hint for (c) (communicated by Hochster): By 6.1.10 we may assume that C is a full subsemigroup of Nm for some m 0, thus C = Nm \ ZC . Set C^ = Nm \ QC . ^ ZC is a torsion group, so that there exist a basis e1 . . . er of ZC^ and Then ZC= positive integers qi for which q1 e1 . . . qr er is a basis of ZC . Extend e1 . . . er to a basis of Qm , and let 'i , i = 1 . . . r, be the linear form on Qm which assigns each vector its i-th coordinate with respect to this basis. Note that 'i(a) 2 Z for all a 2 ZC^ . Then C is the set of elements of c 2 Nm satisfying (i) a system of homogeneous linear equations with rational coe cients whose set of solutions is QC , and secondly the congruence conditions 'i(c) 0 mod qi. Adding positive integral multiples of qi to the coe cients of 'i (with respect to the dual canonical basis of Qm ), we may replace the 'i by linear forms which are non-negative on Nm . Then 'i (c) 0 mod qi if and only if the linear equation 'i (c) = yiqi 6.4.16.
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6.4. Invariants of tori and nite groups
has a non-negative integral solution yi (c). Consider the map C ! Nm Nr , c 7! (c y1 (c) . . . yr (c)). 6.4.17. Let k be an in nite eld, and R = Y1 Y2 Z1 Z2 ]. Suppose that GL(1 k) = k n f0g acts on R by the substitutions Yi 7! aYi, Zi 7! a 1 Zi . (a) Show that S = R G is generated by the elements xij = Yi Zj , i j = 1 2. In Exercise 6.2.7 we have seen that S
= k X11 X12 X21 X22 ]=(X11 X22 ; X21 X12 ), the isomorphism being induced by the substitution Xij 7! xij . (b) Let = (x11 x12 ) and = (x11 x21 ). Show (i) and are prime ideals in S and maximal Cohen{Macaulay S -modules, (ii) j and j are not Cohen{Macaulay for j 2. (4.7.11 is helpful for (ii) use a system of parameters consisting of 1-forms. Or use the Hilbert{Burch theorem.) (c) The characters of GL(1 k) are given by the maps a 7! aj , j 2 Z. Compute the semi-invariants for each of the characters, and nd out which of them are Cohen{Macaulay S -modules. 6.4.18. Let V be a nite dimensional vector space over a eld k, R = S (V ), and G a nite subgroup of GL(V ) such that jGj is invertible in k. Show that for each character of G the R G-module M is a direct R G -summand of R and a rank 1 maximal Cohen{Macaulay R G-module. 6.4.19. Let G be the cyclic subgroup of GL(2 C) generated by the matrix 0 ;1 1 0 and R = S (C2 ) = C X1 X2 ]. Compute the Molien series of G, and show that R G is a complete intersection. (In order to determine the generators of R G one should draw as much information as possible from the Molien series.) 6.4.20. Show that the subgroup of GL(2 C) generated by the matrices 0 ;1 and 0 1 1 ;1 1 0 is isomorphic to S3 , the permutation group of three letters. Prove that R G (with R = S (C2 ) = C X1 X2 ]) is generated by algebraically independent elements x1 x2 , and determine their degrees. 6.4.21. (a) Let k be a eld and S a graded k-algebra generated by elements x1 . . . xn of positive degrees d1 . . . dn . If S is a complete intersection, then there exist positive integers e1 . . . er with r = n ; dim S such that HS (t) = Qr ei Qn dj i=1 (1 ; t )= j =1 (1 ; t ). (b) Embed the group G of the previous problem into GL(4 C) by sending each A 0 matrix A 2 G to the matrix 0 A , and let R = S (C4 ) = C X1 . . . X4 ]. Show that R G is not a complete intersection. Is R G Gorenstein? 6.4.22. Let k be a eld, R = k X1 . . . Xn ], and 1 . . . n be the elementary symmetric polynomials in X1 . . . Xn . (a) Show that height( 1 . . . n )R = n. (b) Let G be the subgroup of GL(n k) formed by the permutation matrices. Noting that G is generated by pseudo-re exions, give a fast proof of the main theorem on symmetric functions in the case in which char k = 0. ;
p
q
p
p
q
q
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6. Semigroup rings and invariant theory
6.5 Invariants of linearly reductive groups Let k be an algebraically closed eld. A linear algebraic group G over k is called linearly reductive if for every nite dimensional representation G ! GL(V ) the k-vector space V splits into the direct sum of irreducible G-subspaces U . Here U is a G-subspace if g(u) 2 U for all g 2 G and u 2 U it is irreducible if it has no G-subspaces other than f0g and U . The main objective of this section is the proof of the following fundamental result. Theorem 6.5.1 (Hochster{Roberts). Let k be an algebraically closed eld and G be a linearly reductive group over k acting linearly on a polynomial ring R = kX1 . . . Xn]. Then the ring R G is Cohen{Macaulay.
The most classical examples of linearly reductive groups are nite groups G whose order is not divisible by char k this is Maschke's theorem. The tori GL(1 k)m are linearly reductive independently of char k, as follows easily from the fact that a torus action can be diagonalized. Thus the results of the previous section about the Cohen{Macaulay property of rings of invariants of tori or nite groups are special cases of 6.5.1. In characteristic 0 the groups GL(n k) and SL(n k) are linearly reductive, and so are the orthogonal and symplectic groups. However, in characteristic p > 0 there exist only a few linearly reductive groups so that 6.5.1 has its main applications in characteristic 0. Let G ! GL(V ) be a nite dimensional representation of a linearly reductive group G. Then the set V G of invariants is the maximal trivial G-subspace of V where trivial means for a G-subspace U that g(u) = u for all g 2 G and u 2 U . Let W be the sum of all non-trivial irreducible G-subspaces of V . Then W is in fact a direct sum W1 Wt of nontrivial irreducible subspaces Wi , and it follows easily that V G \ W = W1G WtG = 0. Thus V = V G W , and W is the unique complementary G G-subspace of V , as every irreducible subspace U with U \ V G = 0 is contained in W . The projection : V ! V G with kernel W satis es the condition (g(v)) = (v) for all g 2 G and v 2 V . It is called the Reynolds operator. As in the previous section let now R = kX1 . . . Xn] be a polynomial ring over k whose space of 1-forms is identi ed with V . Then G acts linearly on the graded components Ri of R for each of which we have a Reynolds operator i . The direct sum of the i is a surjective map : R ! R G which is easily seen to be R G -linear. In fact, the R G -linearity of is equivalent to r Ker Ker for all r 2 R G. It is enough to show that rU Ker for a homogeneous element r 2 R G of degree i and a non-trivial G-subspace U Rj . As multiplication by an invariant is G-linear, rU is either 0 or G-isomorphic to U . Therefore rU Ker i+j . In the general context of linearly reductive groups we have thus recovered
6.5. Invariants of linearly reductive groups
293
the existence of a Reynolds operator R ! R G which was rst encountered above 6.4.4 and which is the crucial fact in the proof of 6.5.1. By 6.4.4 the existence of a Reynolds operator R ! R G implies that IR \ R G = I for every ideal I of R G , and furthermore that R G is a Noetherian k-algebra. Being positively graded, it is even nitely generated over k. So 6.5.1 follows from the more general, purely ring-theoretic Theorem 6.5.2. Let k be a eld, and R = kX1 . . . Xn]. Suppose S is a nitely generated graded k-subalgebra of R such that IR \ S = I for all ideals I of S. Then S is Cohen{Macaulay.
Remarks 6.5.3. (a) The original Hochster{Roberts theorem 201] is more general than stated in 6.5.1. It says: let G be a linearly reductive group
over a eld k acting rationally on a regular Noetherian k-algebra R then R G is Cohen{Macaulay. (b) Let S R be rings S is called a pure subring (or R a pure extension of S ) if for every S -module M the natural homomorphism M = M S S M S R is injective. The reader may prove that S is a pure subring of R if one of the following conditions holds: (i) there exists a Reynolds operator R S (ii) R is faithfully at over S (iii) R=S is at over S . Thus, under the conditions of 6.5.1, R G is a pure subring of R . The choice M = S=I yields that IR S = I for every ideal I of S if S is a pure subring of R . See 270], 7 or Hochster and Roberts 202],
!
!
x
\
Section 5 for a discussion of purity. Using the notion just introduced we can formulate the following even more general theorem of Hochster and Huneke 197]: let R be a regular ring, and S a pure subring of R containing a eld then S is Cohen{Macaulay. We will prove this theorem under the slightly weaker hypothesis that S is a direct S -summand of R see 10.4.1. The case in which R contains a eld of characteristic p > 0 was already given by Hochster and Roberts 201], and the case in which R and S are nitely
generated algebras over a eld was established by Kempf 235]. (c) An important variant of the theorem of Hochster and Roberts is due to Boutot 50]: Let R be a nitely generated algebra over an algebraically closed eld of characteristic 0, and S a pure subring of R if R has rational singularities, then so has S. It is remarkable that Boutot's theorem (for which there is also an analytic version) weakens the hypothesis of the Hochster{Roberts theorem while strengthening its conclusion. For the notion of rational singularity we refer the reader to 236] and 44]. (d) The hypotheses of the theorems presented in (b) and (c) cannot be weakened essentially. In particular it is not true that R G is Cohen{ Macaulay whenever a linearly reductive group G acts linearly on a Cohen{Macaulay ring R . See 6.5.7 for a simple counterexample.
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(e) By 6.1.10 a positive normal semigroup ring S = kC ] can be embedded as a graded subring into a polynomial ring R over k such that there exists a Reynolds operator R ! S . In conjunction with 6.5.2 this argument is the fastest and most elementary proof of the Cohen{Macaulay property of normal semigroup rings, especially if one uses the simple reduction to characteristic p indicated in Exercise 2.1.29. Nevertheless the proof given in 6.3 retains is value since it gave us insight into the combinatorial structure of S , and, above all, allowed us to compute the canonical module. The following proof of 6.5.2 has been drawn from Knop 239]. Its characteristic p part is an argument from tight closure theory, which we will study systematically in Chapter 10. The rst step is the reduction of 6.5.2 to Theorem 6.5.4 below. The k-algebra S is positively graded. By 1.5.17 it has a homogeneous system of parameters f1 . . . fs . Suppose that gr+1fr+1 = g1f1 + + gr fr for some r, 0 r s ; 1. If we can show that gr+1 2 (f1 . . . fr ), then f1 . . . fs is an S -sequence, and the theorem is proved. Suppose on the contrary that gr+1 2= (f1 . . . fr ). As (f1 . . . fr )R \ S = (f1 . . . fr ) by hypothesis, one even has gr+1 2= (f1 . . . fr )R . The elements f1 . . . fs are algebraically independent over k. Moreover, S is a nite kf1 . . . fs ]-module. Therefore it is enough to prove the following theorem. Theorem 6.5.4. Let k be a eld, and f1 . . . fs algebraically independent homogeneous elements of positive degree in R = kX1 . . . Xn]. Suppose that S is a module- nite graded kf1 . . . fs ]-algebra such that there exists a homogeneous homomorphism : S ! R of kf1 . . . fs ]-algebras. If gr+1fr+1 = g1 f1 + + gr fr with g1 . . . gr+1 2 S for some r, 0 r s ; 1, then (gr+1) 2 (f1 . . . fr )R. Proof. Without restriction we may assume that the gi are homogeneous elements of S . Let r1 . . . rm be a system of generators of S as a kf1 . . . fs ]-module. Suppose A is a nitely generated Z-subalgebra of R containing all the elements of k which appear as coecients in (i) (gi ) as a polynomial in X1 . . . Xn, i = 1 . . . r + 1, P (ii) the polynomials piju 2 kY1 . . . Ys ] with ri rj = mu=1 piju(f1 . . . fs )ru , and P (iii) the analogous representations gi = mu=1 qiu(f1 . . . fs )ru . Let B = Af1 . . . fs ], C = AX1 . . . Xn ], and T = B r1 . . . rm ] S . Then Af1 . . . fs ] C (T ) C gi 2 T T = Br1 + Brm : Thus, if we replace k by the nitely generated Z-subalgebra A, then all the assumptions of the theorem (except that on k) remain valid. It is enough to show (gr+1) 2 (f1 . . . fr )C for one such A. As (gr+1) f1 . . . fr are
295
6.5. Invariants of linearly reductive groups
homogeneous this is equivalent to the solubility of a system S of linear equations with coecients in A. (The system S arises from comparing coecients in C = AX1 . . . Xn].) If S has a solution over the eld of fractions of A, then we enlarge A by adjoining the reciprocals of the nitely many denominators of a solution, and obtain a solution in the new A. So suppose that S is insoluble over the eld of fractions of A. Then there is a non-zero element d 2 A such that the reduction of S modulo a maximal ideal of A does not have a solution whenever d 2= . (Simply take d as a suitable subdeterminant of the matrix of S including the right hand side.) Adjoining d ;1 2 k to A, we may assume that the reduction of S modulo any maximal ideal of A is insoluble. We want to pass to such a reduction. It may happen however that the induced map B= B ! C= C is not injective. Therefore an extra condition must rst be satis ed. In fact, by the theorem on generic atness, which we will prove below, there exists t 2 B such that Ct is a free Bt -algebra. As A and B are nitely generated Z-algebras, they are Hilbert rings. This implies that (1) there exists a maximal ideal of B with t 2= , (2) = \ A is a maximal ideal of A, and furthermore (3) A= is a nite eld see A.17, A.18. One has a commutative diagram m
m
m
m
m
n
n
m
n
m
;;;;! C= C ?? ??
B= B m
m
y
y
;;;;! Ct= Ct Since B is a prime ideal with t 2= B , " is injective. Next is injective because the extension Bt ! Ct is faithfully at, and so is injective as
Bt = Bt m
m
m
m
desired. One now replaces all objects by their residue classes modulo . Since the eld A= is nite, the theorem has been reduced to the case in which k is a nite eld! Let p be its characteristic. The nite kf1 . . . fs ]-module S has a rank (just because kf1 . . . fs ] is a domain). Let F be a free submodule of S such that rank F = rank S . There exists a non-zero element c 2 kf1 . . . fs ] such that cS F . We set q = pe , and take the q-th power of the equation gr+1fr+1 = g1f1 + + gr fr and multiply by c to obtain m
m
(cgrq+1)frq+1 =
Xr i=1
(cgiq )fiq :
The elements , i = 1 . . . r + 1 are in the free kf1 . . . fs ]-module F . Then an elementary argument yields hiq 2 F with cgiq = hiq frq+1 for i = 1 . . . r. By substituting these expressions into the previous equation cgiq
296
6. Semigroup rings and invariant theory
and applying : S ! kX1 . . . Xn] one has cfrq+1(gr+1)q =
Xr i=1
fiq frq+1(hiq ) hence c(gr+1)q =
Xr i=1
fiq (hiq ):
Let M be the set of monomials = X1 X1 with i < q for i = 1 . . . n. Taking q-th powers in k is bijective since k is nite. Therefore everyPelement h kX1 . . . Xn] has a necessarily unique representation h = 2M (h )q in particular 1
2
(hiq ) =
Thus
Xr i=1
fiq (hiq ) =
X
2M
X ;Xr 2M i=1
n
(hiq)q :
hiqfi q =
X 2M
(hq)q
with hq 2 (f1 . . . fr )R . The crucialPpoint is that c does not depend on q. We choose q so large that c = 2M c0 with c0 2 k. Let c0 = (c )q . Then
X
2M
(c (gr+1))q =
X
2M
(hq)q :
Since c 6= 0 there exists with c 6= 0, and so 1 (g ) = h 2 (f . . . f )R: r+1
c
q
1
r
A remarkable feature of the preceding proof is that a theorem which has its main applications in characteristic 0 has been reduced to its characteristic p case. Such a reduction will also be fundamental for the results of Chapters 8, 9, and 10. Remark 6.5.5. In view of 6.4.2 and 6.4.9 it is tempting to conjecture that ; R det , if non-zero, is the canonical module of R G under the hypothesis of 6.5.1. Then, in particular, R G would be a Gorenstein ring if det g = 1 for all g 2 G. This was however disproved by Knop 238]: in fact, every ring of invariants R G can be written in the form (R 0)G0 where det g0 = 1 for all g0 2 G0 see Exercise 6.5.8. On the other hand, Knop showed that ; det over an algebraically closed eld of characteristic zero R is indeed the canonical module of R G if the action of G on the vector space V (of 1-forms of R ) satis es a mild non-degeneracy condition the proof uses methods of geometric invariant theory beyond the scope of this book. Knop also proved estimates for the a-invariant a(R G ) in particular one always has a(R G) ; dim R G (compare this with 6.4.2 and 6.4.9). 1
1
297
6.5. Invariants of linearly reductive groups
However, for one class of groups the ring of invariants is always Gorenstein in characteristic 0: if G is semisimple and connected (for example G = SL(n k)), then R G is factorial, and therefore Gorenstein by 6.5.1 and 3.3.19. In fact, let f = 1 m be the prime decomposition of an invariant f . Then the action of an element g 2 G permutes the prime ideals R i. Since fg 2 G : g(R i) = R ig is a non-empty Zariski closed subset of the connected variety G, this set equals G. So g( i ) = i (g) i with i (g) 2 k n f0g i is a character of G. But a semisimple group has no non-trivial characters. Thus g( i ) = i for all g 2 G, and therefore f has a prime decomposition in R G. Generic atness. In the proof of 6.5.4 we used the following theorem on
`generic atness': Theorem 6.5.6. Let R be a Noetherian domain, S a nitely generated Ralgebra, and M a nite S-module. Then there exists f is a free (in particular at) Rf -module.
2 R such that M Rf
Proof. There is nothing to prove for M = 0. So suppose that M 6= 0. Then there exists in M a chain 0 = M0 M1 Mm = M of submodules such that Mi+1=Mi = S= i for some prime ideal i of S . (One only needs that Ass N 6= for an S -module N 6= 0.) It is enough to prove that the theorem holds for each quotient Mi+1=Mi , since N = U N=U if U is a submodule of N for which N=U is free. That is to say, we may suppose that M = S , and, furthermore, that S is a domain. If the natural homomorphism R ! S is not injective, we simply take f from its kernel. Thus R may be considered as a subring of S . Let Q be the eld of fractions of R . Then S Q = SRnf0g is a domain contained in the eld of fractions of S . It is a nitely generated Q-algebra, and therefore has nite Krull dimension, say d . We go by induction on d . By the Noether normalization theorem A.14 the Q-algebra S Q contains y1 . . . yd such that S Q is integral over Qy1 . . . yd ] moreover, y1 . . . yd are algebraically independent over Q. Multiplying by a suitable common denominator, we may assume that yi 2 S for all i. Let z be an element of S . As z is integral over Qy1 . . . yd ], it is easy to nd g 2 R such that z is already integral over Rg y1 . . . yd ]. Since R is a nitely generated R -algebra, one therefore has that Sg is integral over Rg y1 . . . yd ] for some element g 2 R . In view of what is to be proved we may replace R by Rg and S by S Rg . Thus we have reached a situation in which S is a nite module over the ring T = R f1 . . . fd ] S which in turn is isomorphic to a polynomial ring over R , and therefore a free R -module. Let F be a free T -submodule of S such that S=T is a torsion module. Then F is a free R -module. It remains to show that the theorem holds for S=T as a nite p
p
298
6. Semigroup rings and invariant theory
T -module. As above, S=T has a nite ltration with successive quotients of type T = where Spec T . Since S=T is a torsion module, = 0. Therefore T = Q, if non-zero, is a proper residue class ring of T Q, and so has dimension < dim T Q = d . Thus we may repeatedly apply p
p
p
2
p
6
the induction hypothesis in order to complete the proof.
Exercises
Let k be an in nite eld, and let R = k Y1 Y2 Z1 Z2 ]=(Y12 ; Y22 ). Obviously R is a Cohen{Macaulay ring, and reduced if char k 6= 2. Let G = GL(1 k) act on k Y1 Y2 Z1 Z2 ] by the substitutions Yi 7! aYi , Zi 7! a 1 Zi , a 2 k, a 6= 0. Prove: (a) The action of G induces an action of G on R , and R G is the k-subalgebra generated by the products yi zj , i j = 1 2. (Small letters denote residue classes in R .) (b) The substitution Xij 7! YiZj induces a surjective k-algebra homomorphism k X11 X12 X21 X22 ] ! R G . Its kernel is generated by the elements X11 X22 ;X12 X21 , X112 ; X122 , X11 X21 ; X12 X22 , X212 ; X222 . (c) R G is not Cohen{Macaulay. By increasing the number of variables Yi (and the degree of the equation de ning R ) one can even produce examples of factorial hypersurface rings R such that R G is not Cohen{Macaulay. (A hypersurface ring is a residue class ring of a regular ring with respect to a principal ideal.) 6.5.8. Suppose G be a subgroup of GL(V ) where V is a nite dimensional vector space over an in nite eld k. Set V = k2 k3 V and let G = SL(2 k)SL(3 k)G act on V by 6.5.7.
;
0
0
0
(f h g)(u w v) = (det g)f (u) (det g) 1 h(w) g(v) : ;
Then obviously det g = 1 for all g RG
= (R )G0 . 0
0
0
2 G . Let R = S (V ) and R 0
0
= S (V ). Show 0
Notes Hochster 174] proved the Cohen{Macaulay property of normal semigroup rings using the shellability of convex polytopes (see Section 5.2 for the notion of shellability.) A purely algebraic proof was provided by Goto and Watanabe 135] they computed local cohomology from a complex similar to L . Such complexes, or their graded k-duals (which are dualizing complexes) have been constructed by several authors. See Trung and Hoa 371] or Schafer and Schenzel 321] these articles give Cohen{ Macaulay criteria for general ane semigroup rings. For a general ane semigroup C the Cohen{Macaulay and the Gorenstein property of kC ] may depend on the eld k see 371] and Hoa 173]. Gilmer 126] treats semigroup rings from a more general point of view. Our approach is close to that of Ishida 227] from Danilov 76] we borrowed the idea of proving the vanishing of certain cohomology groups .
Notes
299
by a topological argument. See also Stanley 360] and 363] where the method is also applied to certain modules over normal semigroup rings, namely those which in the invariant-theoretic situation arise as semiinvariants. (See 6.4.17 for a non{Cohen{Macaulay such module.) More recently the Cohen{Macaulay property of modules of semi-invariants was investigated by Van den Bergh 378]. The complete intersection normal semigroup rings were classi ed by Nakajima 285]. Stanley computed the canonical module of a normal semigroup ring by a combinatorial argument outlined in 6.3.10, whereas Danilov 76] applied di erentials. Local cohomology as in 6.3.5 was used by Goto and Watanabe. For the theory of Ehrhart polynomials and related combinatorial functions we refer the reader to Ehrhart 89], Stanley 361], and to Hibi's survey 168] where numerous references are given. See also Danilov 76]. Normal semigroup rings, or rather their spectra, are the most special cases of toric varieties which connect combinatorics and algebraic geometry. We must con ne ourselves to a list of references: Kempf, Knudsen, Mumford, and Saint{Donat 236], Danilov 76], Oda 293], and Ewald 99]. The recent book by Sturmfels 366] treats the combinatorial aspects of Grobner bases for the de ning ideals of semigroup rings. The invariant theory of nite groups is a classical subject whose literature we cannot cover adequately instead we refer the reader to Springer 355], Stanley 358], Benson 40], and Smith 353]. While 6.4.5 is due to Hochster and Eagon 189], the Cohen{Macaulay property of rings of invariants of nite groups seems to have been realized by several authors. The characterization of Gorenstein invariants is the work of Watanabe as pointed out in 6.4.11 Stanley gave the combinatorial proof reproduced by us. The determination of the canonical module is only implicit in Watanabe's papers according to Stanley 358] it was made explicit by Eisenbud. References for the Hochster{Roberts theorem, its variants and extensions have been indicated in 6.5.3. Hochster 187] contains an extensive discussion of the problem of determining the canonical module of a ring of invariants. As pointed out in 6.5.5, this problem was satisfactorily solved by Knop 238]. The example in 6.5.7 is a simpli cation of that of Hochster and Eagon 189], p. 1056. It is a very special instance of the Segre product of graded rings. The Cohen{Macaulay property of Segre products was explored by Chow 69] and Goto and Watanabe 134]. Hochster 185] is a survey of the invariant theory of commutative rings. The theorem of generic atness is due to Grothendieck 142], IV.6.9. A more re ned version was given by Hochster and Roberts 201] see also 270], x24.
7 Determinantal rings
Determinantal rings occur in algebraic geometry as coordinate rings of classical algebraic varieties. From the algebraic point of view they are graded algebras with straightening law which themselves form a subclass of the class of graded Hodge algebras. The special feature of such an algebra is that it is free over the ground ring with a monomial basis whose multiplication table is compatible with a partial order on the algebra generators. The results on ltered rings in Section 4.5 will be applied to `trivialize' a graded Hodge algebra: by repeatedly passing to a suitable associated graded ring one eventually gets a discrete Hodge algebra, which is nothing but the residue class ring of a polynomial ring modulo an ideal generated by monomials. A discrete algebra with straightening law may be considered the Stanley{Reisner ring of the order complex of a certain poset, and as an application we will thus obtain a Cohen{Macaulay criterion. The remaining sections of the chapter are devoted to the most important examples of algebras with straightening law, the determinantal rings. It will be shown that these rings are normal Cohen{Macaulay domains. The class group and the canonical module will be identi ed, and we will characterize the Gorenstein determinantal rings. 7.1 Graded Hodge algebras In this section we introduce graded Hodge algebras and study their basic properties. Let A be aQring, H a nite subset of A, and c 2 NH , c = (c ). An element u = 2H c is called a monomial on H with exponent c. Its support is the set supp u = f 2 H : c 6= 0g. Let u and u0 be monomials on H with exponents c and c0 , respectively. We say u divides u0 or u is a factor of u0 if c0 ; c 2 NH . Finally, if NH is a semigroup ideal, we call c 2 a generator of if c ; c0 62 NH for all c0 2 , c0 6= c. De nition 7.1.1. Let A be a B -algebra, H A a nite subset with partial order , and NH a semigroup ideal. A is a graded Hodge algebra on H over B L governed by if the following conditions hold. (H0) A = i 0 Ai is a graded B algebra with A0 = B , and H consists of elements of positive degree and generates A over B . 300
301
7.1. Graded Hodge algebras
(H1) The monomials on H with exponent in NH n are linearly independent over B . They are called standard monomials. (H2) (Straightening law) If v is a monomial on H whose exponent is a generator of , then v has a presentation v=
X
buu
bu
2 B
bu = 0 u a standard monomial,
6
bu
2 B
bu = 0 u a standard monomial,
such that for each 2 H which divides v there exists for every u a factor 'u with 'u < . The right hand side of a straightening relation may of course be the empty sum, i.e. equal to zero. If this happens for all straightening relations, the graded Hodge algebra is called discrete. In Qthis case A = B X : 2 H ]=I where I is generated by the monomials 2H Xc , c = (c ), c 2 . In particular, Stanley{Reisner rings are discrete Hodge algebras. The graded Hodge algebra A is called a graded algebra with straightening law (on H over B ), abbreviated graded ASL, if is generated by the exponents of monomials ( where and ( are incomparable elements in H . It follows that a monomial u is standard if and only if all factors of u are comparable with each other, and for all incomparable ( 2 H we have a straightening relation (=
X
bu u
6
satisfying the condition: every u contains a factor ' 2 H such that ' < , ' < (. In fact, by (H2 ) there exist factors 'u and 'u0 of u such that 'u < and 'u0 < (. Since all factors of u are comparable with each other we may choose for ' the minimum of 'u and 'u0 . ASLs are the most important graded Hodge algebras. Signi cant examples will be treated in the next sections. Proposition 7.1.2. Let A be a graded Hodge algebra on H over B governed by . Then the standard monomials form a B-basis of A. Proof. Let H we de ne dim to be the maximal length of Q chains = P in H , and de ne the weight of a monomial u = P2H c 0< 1< to be 2H c (d +1)dim , where d is the maximum of the numbers 2H c of generators c = (c ) of .
2
It suces to show that all non-standard monomials are linear combinations of standard monomials. Let v0 be a monomial with exponent c0 2 Q , and let c be a generator of such that c0 ; c 2 NH . Then v = 2H c divides v0 , and so v0 = vw where w is a monomialPon H . Applying the straightening law for v, we obtain the equation v0 = buuw, bu 2 B , bu 6= 0, u standard. We claim that all monomials on the right hand side of this equation are of strictly greater weight than v0 . In
302
7. Determinantal rings
fact, if = maxfdim : c 6= 0g, then for any u in the straightening equation P for v there exists a factor u with < dim u, so that weight v 2H c (d + 1) < (d + 1) +1 (d + 1)dim u weight u. Since the weight of a product of monomials is the sum of the weights of the factors, the claim follows. On the other hand, the monomials on the right hand side of the equation for v0 have the same degree as v0 . Therefore descending induction concludes the proof. The previous proposition guarantees that every element of A has a unique presentation as a B -linear combination of standard monomials, which we call its standard representation. Proposition 7.1.3. Let A be a graded Hodge algebra on H over B governed
2
by , and T , H, a set of indeterminates over B. For each monomial u= 1 2 on H we set Tu = T Tn . Then the kernel of the Bn algebra epimorphism
1
2 H ] ;! A T 7! P is generated by the elements Tv ; bu Tu corresponding to the straightening relations. Proof. P Let I be the ideal in BT : 2 H ] generated by the elements Tv ; buTu corresponding to the straightening relations. It is clear that I Ker '. Conversely, let f 2 Ker ' then P the proof of 7.1.2 shows that there exists g 2 I Psuch that f ; g = buTu , u standard. It follows that 0 = '(f ; g) = buu. According to (H1 ) all bu = 0, and hence f 2 I. ' : B T :
Among the graded Hodge algebras on H over B governed by , the discrete Hodge algebra is in a sense the simplest. Its ring-theoretic properties are determined only by the ground ring B and the combinatorial properties of H and . Surprisingly this is true in part for a general graded Hodge algebra as well. The set Ind A of elements 2 H which appear as factors in the monomials on the right-hand side of the straightening relations is called the indiscrete part of A. It serves as a measure of how much A di ers from a discrete Hodge algebra. The following theorem allows the stepwise approach from a general graded Hodge algebra to a discrete one by forming suitable associated graded rings. The results of 4.4 permit us to control ring-theoretic properties of the algebras involved in this operation. Suppose Ind A 6= , choose a minimal element 0 2 Ind A, and set I = ( 0). We will rst prove a re nement of 7.1.2. Lemma 7.1.4. The ideal I j has a B-basis consisting of all standard monQ omials u = 2H c such that c j. 0
7.1. Graded Hodge algebras
303
Certainly the elements 0j u, u a standard monomial, generate I j as a B -module. We claim that 0j u either is a standard monomial or is zero. This, in view of 7.1.2, will prove the lemma. Suppose 0j u is not standard then it is a multiple of a monomial v whose exponent is a generator of . Since u is standard, the element 0 is a factor of v, and thus for each monomial on the right hand side of the straightening relation for v there exists a factor less than 0 . Since 0 is a minimal element among such factors, the straightening relation must be trivial. It follows that 0j u = 0. For an element a 2 A we de ne ord a as the supremum of integers j for which a 2 I j , and call a? = a + I ord a+1 the initial form of a in grI (A). We have ord 0 = 1, and ord = 0 for all 2 H , 6= 0. Let a b 2 A then ord ab ord a + ord b, and (ab)? = a? b? if ord ab = ord a + ord b. By induction oneQproves a similar formula for more than just two factors. Thus, if u = 2H c is aQstandard monomial in A, it follows from the previous lemma that u? = 2H ( ?)c . In conclusion we see that grI (A) is generated over B by the elements ? , 2 H , and that grI (A) is a free B -module with basis fu? : u is a standard monomial of Ag. Moreover, grI (A) may be viewed as a positively graded B -algebra, if, for all j 0, the set of homogeneous elements of degree j of grI (A) is de ned to be fa? : a 2 A is homogeneous of degree j g. Now it is easy to give grI (A) the structure of a graded Hodge algebra: we let H ? = f ? : 2 H g. The map H ! H ? , 7! ? , induces a bijection ' : NH ! NH ? , and we set ? = '( ). The partial order de ned on H ? will of course be given by ? < !? () < !. Theorem 7.1.5. grI (A) is a graded Hodge algebra on H ? over B governed by ? , and Ind grI (A) f ? : 2 Ind Ag n f 0? g. Proof. It remains to check (H2 ): let d = (d? ) be a generator of ? , and Q w = ?2H ? ( ? )d?Q . Then c = (c ), c = d? for all 2 H , is a generator P of , and for v = 2H c we have the straightening relation v = buu. We want to de ne the straightening relation for w. There are two cases to consider. In the rst case, 0 is a factor of v. Then v = 0v0 where v0 is a standard monomial, and it follows that v = 0 as we saw in the proof of 7.1.4. Therefore w = 0 is the desired straightening relation P in this case. In the second case, 0 is not a factor of v. Then w = ord u=0 buu? is the straightening relation for w. In particular it follows that 0? 62 Ind grI (A). Proof.
Let A (respectively A0) be a graded Hodge algebra on H (respectively over B governed by (respectively 0 ). We say that A and A0 are Hodge algebras with the same data, if there is an isomorphism H ! H 0 of posets for which the corresponding map NH ! NH 0 induces a bijection H 0)
304
7. Determinantal rings
! 0. We have just seen that A and grI (A) are graded Hodge algebras
with the same data, the only di erence being that the indiscrete part has become smaller. Thus after nitely many such steps we arrive at a discrete Hodge algebra with the same data as A. Corollary 7.1.6. Let A be a graded Hodge algebra on H over the Noetherian
ring B governed by , and let A0 be the discrete Hodge algebra over B with the same data. Then: (a) dim A = dim A0 (b) A is reduced, Cohen{Macaulay, or Gorenstein if A0 is too. Proof. A basic observation for (a) and (b) is that both A and A0 are free and, hence, faithfully at B -algebras. In conjunction with A.11 it implies
dim A = max(dim B + dim A k( )) where ranges over Spec B . It is clear that A k( ) is a Hodge algebra over k( ) with the same data as A therefore A0 k( ) = (A k( ))0 . Thus it is enough to consider the case in which B = k is a eld. Next we may replace A0 by grI (A). Both these rings are positively graded so that dim A = dim A and dim A0 = dim A where and 0 are the maximal ideals. Furthermore I is generated by homogeneous elements of positive degrees. Therefore 4.5.6 yields the desired equality of dimensions. We show (b) for the Cohen{Macaulay property. The Cohen{Macaulay property of A0 implies that of B and that of A0 k( ) for all prime ideals of B (see 2.1.7). Let be a prime ideal of A and set = B \ . Then depth A = depth B + depth A k( ) = dim B + depth(A k( )) where is the image of in A k( ) (see 1.2.16). It follows that A is Cohen{Macaulay if depth(A k( )) = dim(A k( )) . Thus the isomorphism A0 k( ) = (A k( ))0 reduces the contention once more to the case in which B = k is a eld. It is enough to derive the Cohen{Macaulay property of A from that of grI (A). Let be the maximal ideal of A. Since I , 4.5.7 implies that A is Cohen{Macaulay, and then A is, too, by Exercise 2.1.27. For the Gorenstein property one argues similarly, using 3.3.15, 4.5.7, and Exercise 3.6.20. The assertion about A being reduced follows immediately from 4.5.8. In case A is an ASL on H over B , the discrete ASL with these data is the Stanley{Reisner ring of the order complex (H ) (see Section 5.1) over B . Thus we may use the results of Chapter 5 in order to conclude that certain ASLs are Cohen{Macaulay. In the following corollary we extend the poset H by adding absolutely minimal and maximal elements 0^ and 1^ . p
p
p
p
p
p
m
m
0
m
p
m
p
p
P
p
P
p
P
p
P
p
p
q
P
q
p
p
P
p
q
p
m
m
q
p
m
305
7.1. Graded Hodge algebras
Corollary 7.1.7. Let A be an ASL on H over B. If B is Cohen{Macaulay and H f0^ 1^ g is a locally upper semimodular poset, then A is Cohen{ Macaulay .
By virtue of 5.1.12 and 5.1.14 the discrete ASL A00 on H f0^ 1^ g is Cohen{Macaulay, provided B is a eld. Since A00 is obviously a polynomial ring over the discrete ASL A0 on H , it follows that A0 is Cohen{Macaulay. According to Exercise 5.1.25, A0 is Cohen{Macaulay for every Cohen{Macaulay ring B . Now one applies 7.1.6. Proof.
There is a simple proof of 7.1.7 which avoids the combinatorially dicult theorem 5.1.12 see 90] or 61], (5.14). Exercises Let A be a graded Hodge algebra on H over a ring B governed by . (a) Let H be a subset of H such that the ideal H A is generated as a B -module by the standard monomials it contains. Show that A=H A is in a natural way a Hodge algebra on H n H governed by where H n H is considered as a subposet of H and consists of all elements of which are exponents of monomials on H nH . (b) Show that (a) in particular applies if H is an ideal in H . (An ideal in H is a subset satisfying the following condition: h h 2 H ) h 2 H .) (c) Let H and H be ideals in H . Then H A \ H A is the ideal of A generated by H \ H . (d) Specialize (a), (b), and (c) to the case of an ASL A. 7.1.9. With the notation of 7.1.8 assume that is generated by squarefree monomials. Show that A is reduced if (and only if) B is reduced. In particular a graded ASL over a reduced ring B is reduced. 7.1.10. Let A be a graded ASL on the poset H over a Noetherian ring B . Show that dim A = dim B + rank H + 1. Hint: First prove the formula for a eld B . Next deduce dim A = maxfdim B + dim k( ) A : 2 Spec B g from A.11 and the fact that A is a free B -module, and note that k( ) A is an ASL on H over the eld k( ). 7.1.11. (Hibi) Let k be a eld, C a positive a ne semigroup generated by c1 . . . cn , and A = k C ]. We order fc1 . . . cn g by setting c1 < < cn , and let be the set of exponents (a1 . . . an ) such that there exists (b1 . . . bn ) which is lexicographically greater than (a1 . . . an ) and satis es the condition (X c )a (X cn )an = (X c )b (X cn )bn . (a) Show k C ] is a graded Hodge algebra over k with these data. (b) Let C be the subsemigroup of N2 generated by (2 0), (2 1), (1 2), and (0 2). Determine the sets for the orders (i) (2 0) < (2 1) < (1 2) < (0 2) and (ii) (2 1) < (1 2) < (2 0) < (0 2), and show that the discrete Hodge algebra A0 is Gorenstein in case (i), and not even Cohen{Macaulay in case (ii). 7.1.8.
0
0
0
0
0
0
0
0
0
0
0
0
0
00
0
0
0
0
00
00
p
p
p
p
1
1
p
1
1
306
7. Determinantal rings
7.2 Straightening laws on posets of minors The most important examples of ASLs are rings related to matrices and determinants, and the prototype of such a ring is Rr+1 = Rr+1(X ) = B X ]=Ir+1(X )
where B X ] is the polynomial ring in the entries of an m n matrix of indeterminates Xij over some ring B of coecients, and Ir+1(X ) denotes the ideal generated by the (r + 1)-minors of X . We always suppose that 0 r min(m n), the trivial cases r = 0 and r = min(m n) being included for reasons of systematics. What makes the analysis of Rr+1 dicult is the fact that the generators of Ir+1(X ) are very complicated expressions in terms of the Xij . Therefore one enlarges the set of generators of the B -algebra B X ] by considering each minor as a generator. Of course, apart from trivial cases, we lose the algebraic independence of the generating set, but only to the extent that B X ] is an ASL on the set of minors of X . The minor corresponding to the submatrix of X with rows a1 . . . au and columns b1 . . . bu is denoted by a1 . . . au j b1 . . . bu ]: The set consists of those minors a1 . . . au j b1 . . . bu ] which satisfy the condition a1 < < au, b1 < < bu. It is partially ordered by the rule a1 . . . au j b1 . . . bu] c1 . . . cv j d1 . . . dv ] () u v and ai ci bi di i = 1 . . . v: It is easy to see that is a distributive lattice under this partial order. Rather than proving directly that B X ] is a graded ASL on we take a detour which leads to a substantial simpli cation of the combinatorial details, and introduces another interesting and important class of rings. We suppose that m n. Then the maximal minors of X are the m-minors. An m-minor of X is simply denoted by a1 . . . am] where a1 . . . am are the column indices of the submatrix whose determinant is taken. The subset of consisting of all m-minors in is called ; . Obviously ; is a sublattice of . We write G(X )
or, if appropriate, GB (X ) for the B -subalgebra of B X ] generated by ; . The letter G has been chosen since (over a eld B ) G(X ) is the coordinate ring of the Grassmannian of m-dimensional vector subspaces of B n.
307
7.2. Straightening laws on posets of minors
Theorem 7.2.1 (Hodge). Let B be a ring, and X an m n matrix of indeterminates over B with m n. Then G(X ) is a graded ASL on ; . Condition (H0 ) is evidently satis ed, and the validity of (H1 ) is stated in the following lemma: Lemma 7.2.2. The standard monomials in ; are linearly independent. Proof. For = a1 . . . am ] let U be the following m n matrix whose entries Uij are indeterminates over B : 0 0 0 U1a U1a ;1 U1a U1a ;1 U1am U1n 1 0 0 U2a U2a ;1 C BB .. .. C BB CC : 0 0 . . C B@ .. .. .. .. .. .. A . . . . . . 0 0 0 0 0 0 Umam Umn The substitution which maps Xij to the corresponding entry of U induces a B -algebra homomorphism ' : G(X ) ! G(U ) where G(U ) denotes the B -subalgebra of B Uij : j ai] generated by the maximal minors of U . Observe that for the matrix U has indeterminate entries where U has non-zero entries. Therefore the analogous substitution yields a B -algebra homomorphism : G(U ) ! G(U ) with ' = ' . The matrix U is chosen in such a way that the submatrix of its columns 1 . . . ai ; 1 has rank i ; 1 for i = 1 . . . m + 1 (where we let am+1 = n + 1). The reader may carefully check that this implies ' ( ) = 0 for all 6 . So the application of ' to a linear combination of standard monomials strips o all terms which contain a factor 6 . The lemma follows immediately from the following claim (with = 1 . . . m]): let () be the set of standard monomials all of whose factors are . then ' ( ()) is a linearly independent subset of G(U ). this claim by descending induction over the poset ; . Let P Webuprove u2U ' (u) = 0 be a linear combination with U (), U 6= , and bu 6= 0 for all u 2 U . The element ' () is a product of indeterminates, and therefore G(U )-regular. Thus, cancelling ' () if necessary, we may suppose that does not occur as a factor of at least one of the standard monomials in the sum, say u0. Let 0 be P P the smallest factor 0 of u0. Then > , and 0 = u2U bu 0 (' (u)) = u02U 0 bu0 ' 0 (u0 ) where U 0 = U \ (0). Since u0 2 U 0 , we obtain a contradiction to the induction hypothesis. Next we want to show that every product of (incomparable) minors 2 ; can be written as a linear combination of standard monomials. This `straightening' will be performed by iterated applications of the Plucker relations given in the following lemma. (We use (i1 . . . is ) to denote the sign of the permutation of f1 . . . sg represented by the sequence i1 . . . is .) 1
2
2
3
2
3
308
7. Determinantal rings
Lemma 7.2.3. For every m n-matrix, m n, with elements in a ring A and all indices a1 . . . ap , bq . . . bm , c1 . . . cs 2 f1 . . . ng such that s = m ; p + q ; 1 > m and t = m ; p > 0 one has
X
i1 < bp+1. We put q = p + 2, s = m + 1, (c1 . . . cs ) = (ap+1 . . . am b1 . . . bp+1). Then, in the Plucker relation 7.2.3 with these data, all the non-zero terms d1 . . . dm ]e1 . . . em ] 6= a1 . . . am ]b1 . . . bm ] have the following properties (after their column indices have been arranged in ascending order): (a) d1 . . . dm] a1 . . . am] (b) d1 e1 . . . dp+1 ep+1: Proof. Since b1 < < bp+1 < ap+1 < < am , d1 . . . dm ] arises from a1 . . . am ] by a replacement of some of the ai by smaller indices. This implies (a) and di ei for i = 1 . . . p. Furthermore dp+1 2 fa1 . . . ap b1 . . . bp+1g, so dp+1 bp+1, and ep+1 2 fap+1 . . . am bp+1 . . . bmg, so bp+1 ep+1. By induction on p it follows immediately from 7.2.4 that a product , 2 ; , is a B -linear combination of standard monomials , , such that . This however does not yet imply (H2). (In general the straightening procedure based on 7.2.4 produces intermediate results violating (H2 ).) In order to see that (H2 ) is indeed satis ed we must also straighten the product in the order . The standard monomials obtained now satisfy . By linear independence of the standard monomials, both representations of = coincide, and (H2) follows. This completes the proof of 7.2.1. Before we turn to the discussion of the polynomial ring B X ], we state a useful corollary of 7.2.1. Corollary 7.2.5. (a) Let T , 2 ; , be a set of indeterminates over B. Then the kernel of the surjective homomorphism B T : 2 ; ] ! G(X ) is generated by the elements representing the Plucker relations. (b) One has GB (X ) = GZ (X ) B. (c) Suppose B is a Noetherian ring. Then dim G(X ) = dim B + m(n m)+1. Proof. By 7.1.3, GB (X ) is the residue class ring of B T : ; ] modulo
2
;
the ideal generated by the elements representing the straightening relations. As seen above, the straightening relations are linear combinations of the Plucker relations. This proves (a), and (b) is a simple consequence of (a) if one notes that the Plucker relations are de ned over Z. By virtue of 7.1.10 one has dim G(X ) = dim B +rank ; +1. Each cover of a1 . . . am ] 2 ; is obtained by replacing one of the indices ai by ai +1 (of course, this is only feasible if aPi +1 < ai+1). Furthermore rank1 . . . m] = 0. P Therefore ranka1 . . . am ] = mi=1 ai ; i = mi=1 ai ; m(m + 1)=2. This immediately yields rank ; = rankn ; m + 1 . . . n] = m(n ; m).
310
7. Determinantal rings
As before let X be a matrix of indeterminates, and the poset of its minors the condition m n is no longer required. We extend X by m columns of further indeterminates, obtaining
0 X11 X0 = B @ ... Xm1
X1n X1n+1
X1n+m
Xmn Xmn+1
Xmn+m
.. .
.. .
.. .
1 CA :
Then B X 0] is mapped onto B X ] by substituting for each entry of X 0 the corresponding entry of the matrix 0 X11 X1n 0 0 1 1 .. BB ... ... 0 C C . BB .. .. .. . . . . . . .. C . . . . . .C CC : BB . . . . .. A @ 0 .. .. Xm1 Xmn 1 0 0 Let ' : G(X 0) ! B X ] be the induced homomorphism. Then (1) '(b1 . . . bm ]) = a1 . . . at j b1 . . . bt] where t = maxfi : bi ng and a1 . . . at have been chosen such that fa1 . . . at n + m + 1 ; bm . . . n + m + 1 ; bt+1g = f1 . . . mg: Equation (1) shows that ' is surjective, and furthermore sets up a bijective correspondence between the set ; 0 of m-minors of X 0 and f1g. Note that the maximal element e = n + 1 . . . n + m] of ; 0 is mapped to 1, and that the restriction of ' to ; 0 nfeg is an isomorphism of posets. (We leave the veri cation of this fact to the reader the details can also be found in 61], (4.9).) Lemma 7.2.6. The kernel of ' : G(X 0) ! B X ] is generated by e 1. Proof. Note that G(X 0), B X ], and ' are obtained from the corresponding objects over Z by taking tensor products of the latter with B . (This is non-trivial only for G(X 0) for which it has been stated in 7.2.5.) Therefore it is sucient to consider the case B = Z. Then G(X 0) is an integral domain, and it follows from the properties of dehomogenization (see 1.5.18) that e 1 generates a prime ideal of height 1. By virtue of 7.2.5 one has dim G(X 0) = dim Z + mn + 1 = dim ZX ] + 1: As Ker ', we in fact have = Ker '. Theorem 7.2.7 (Doubilet{Rota{Stein). Let B be a ring, and X an m n matrix of indeterminates. Then B X ] is a graded ASL on the poset of p
p
minors of X.
p
311
7.3. Properties of determinantal rings
From our previous arguments it is obvious that ' maps the standard monomials in ; 0 n feg to standard monomials in . Since ' is surjective and G(X 0) is a graded ASL on ; 0 , the standard monomials in generate B X ] as a B -module. The smallest factor of a standard monomial on the right hand side of a straightening relation in an arbitrary ASL is never a maximal element of the underlying poset. Therefore the validity of (H2) cannot be destroyed by a substitution which takes such an element to an element of B . It only remains to observe the linear independence of the standard monomials in , or, equivalently, that there is no non-trivial relation Proof.
X
au u = (e
1) X bv v
where u and v represent pairwise distinct standard monomials in G(X 0), and none of the u contains e as a factor. Exercises
Let u = 1 r be a product of minors of X . The content of u is the vector of length m + n which for each row and each column lists the multiplicity with which the row or column appears in u. Let v be a standard monomial in the standard representation of u. Show that v has the same content as u, and has at most r factors. 7.2.9. Let m n, and X be an m n matrix of indeterminates over a ring B . Let 1 r1 < < rs m be integers, and consider the subposet ; (r1 . . . rs ) of formed by all minors which are of the form 1 . . . ri j a1 . . . ari ] for some i. Show that B ; (r1 . . . rs )] is a graded ASL on ; (r1 . . . rs ). (Note that this class of rings generalizes G(X ).) (For a eld B = k, k ; (r1 . . . rs )] is the multihomogeneous coordinate ring of the variety of ags 0 U1 Ur kn of linear subspaces such that dim Ui = ri .) 7.2.10. Let H be a nite poset with partial order , and H the poset with the reverse partial order : h h () h h. A graded ASL R on H is called symmetric if it is also an ASL on H (with respect to the same embedding H ! R ). (a) Show that G(X ) is a symmetric ASL. (b) Show that the graded ASLs B ; (r1 . . . rs )] of the previous exercise are symmetric. 7.2.8.
;
0
0
;
7.3 Properties of determinantal rings In this section we shall assume that the ground ring B is a eld, and therefore replace the letter B by k throughout. The transfer of the results to more general ground rings is indicated in Exercise 7.3.8. As in the previous section let X be an m n matrix of indeterminates over k. The determinantal ring Rr+1 is the residue class ring of kX ] with respect to the ideal generated by the (r + 1)-minors of X . In view of the
312
7. Determinantal rings
ASL structure of kX ] it is useful to extend this system of generators by including all t-minors with t r +1. The enlarged system of generators is an ideal in the poset of all minors of X . By Exercise 7.1.8, Rr+1 inherits the ASL property of kX ] its underlying poset is the coideal r+1 of which consists of the u-minors of X with u < r + 1. (A coideal is the complement of an ideal.) Evidently r+1 has a single minimal element, namely 1 . . . r j 1 . . . r]. More generally, for 2 we want to investigate the residue class rings R of kX ] modulo the ideal I generated by all minors " 6 . As in the special case = 1 . . . r j 1 . . . r] above, R is an ASL on the coideal = f' 2 : ' g. In a distributive lattice a coideal with a single minimal element is again a distributive lattice, and it follows directly from 7.1.7 and 7.1.9 that R is a reduced Cohen{Macaulay ring. Moreover we can easily compute its dimension. Similarly we may consider the residue class rings G of G(X ). These are the residue class rings of G(X ) with respect to the ideal J generated by all 2 ; , 6 . The corresponding coideal in ; is denoted ; . Theorem 7.3.1. Let k be a eld, and X an m n matrix of indeterminates over k.
(a) (Hochster, Laksov, Musili) Suppose m n, and let = a1 . . . am ] 2 ; . Then G is a normal Cohen{Macaulay domain of dimension
; m) + m(m2+ 1) ; X ai + 1: i=1 (b) Let = a1 . . . ar j b1 . . . br ] 2 . Then R is a normal Cohen{Macaulay m
m(n
domain of dimension
(m + n + 1)r ;
Xr i=1
(ai + bi ):
(c) (Hochster{Eagon) In particular, Rr+1 is a normal Cohen{Macaulay domain of dimension (m + n ; r)r. Proof. The Cohen{Macaulay property of R and G follows from the fact that the posets and ; are distributive lattices, as explained above. In order to compute their dimensions one must determine the ranks of the posets and ; see 7.1.10. Since all maximal chains in a distributive lattice have the same length, one has rank ; = rank ; ; (rank + 1): Both rank ; and rank were computed in the proof of 7.2.5. For the computation of dim R and for the proof of normality it is convenient to relate R to a ring of type G 0 in the same way as B X ]
313
7.3. Properties of determinantal rings
was related to G(X 0) for the proof of 7.2.7. We choose 0 = b1 . . . br (n + m + 1) ; a01 . . . (n + m + 1) ; a0m;r ] with fa01 . . . a0m;r g being the complement of fa1 . . . ar g in f1 . . . mg. Then ' : G(X 0) ! B X ] as de ned before 7.2.6 maps 0 to and the generators of J 0 to a generating set of I . It follows from 7.2.6 that the induced homomorphism ' : G 0 ! R is surjective, and that its kernel is generated by e 1 where the maximal element e of ; 0 is considered as an element of G 0 . Now an easy computation yields the dimension of R . As we just saw, R is a dehomogenization of a ring of type ; , and therefore it is sucient to prove that G is a normal domain (see Exercise 2.2.34). Note rst that G is indeed a domain: the surjective homomorphism ' : G(X ) ! G(U ) constructed in the proof of 7.2.2 induces a homomorphism ' : G ! G(U ), and ' maps the standard basis of G onto a linearly independent subset of G(U ) as was shown there. So G is isomorphic to the integral domain G(U ). To prove normality we apply the criterion in Exercise 2.2.33 with x = : being the single minimal element of ; , is evidently ; -regular moreover, G =() is an ASL on ; n fg, and therefore reduced. Finally 7.3.2 shows that G ;1 ] is a normal domain. Theorem 7.3.1(c) entails the classical formula height Ir+1(X ) = (m ; r)(n ; r): Thus Ir+1(X ) has maximal height: by a theorem of Eagon and Northcott one has height Ir+1(x) (m ; r)(n ; r) for an arbitrary m n matrix x over a Noetherian ring S , provided Ir+1(x) 6= S see 61], (2.1) or 270], x13. Lemma 7.3.2. With the notation of 7.3.1 let = d1 . . . dm ] 2 ; : ai 2= d1 . . . dm] for at most one index i : Then
G ;1 ] = k ;1 ] and the elements of are algebraically independent over k. In particular G ;1 ] is a regular domain. Proof. We show that e1 . . . em ] k ;1 ] for all e1 . . . em ] ; by induction on the number w of indices i such that ei = a1 . . . am]. For w = 0 and w = 1, e1 . . . em ] by de nition. Let w > 1 and choose an index j such that ej = a1 . . . am ]. We use the Plucker relation 7.2.3, the
2
2
2
2
2
data of 7.2.3 corresponding to the present ones in the following manner: p = 0, q = 2, s = m + 1, (b2 . . . bm ) = (e1 . . . ej ;1 ej +1 . . . em ), and (c1 . . . cs ) = (a1 . . . am ej ). In this relation all the terms di erent from a1 . . . am]ej e1 . . . ej ;1 ej +1 . . . em ] = (;1)j ;1a1 . . . am ]e1 . . . em ]
314
7. Determinantal rings
and non-zero in G have the form " such that 2 and " has only w ; 1 indices not occurring in . Solving for a1 . . . am ]e1 . . . em ] and dividing by , one gets e1 . . . em ] 2 k ;1 ]. For the proof of the algebraic independence of we rst note that dim G ;1 ] = dim G . This follows easily from A.16 if one uses that G is an ane domain over k as was demonstrated in the proof of 7.3.1. Now it is enough to verify that j j = dim G , a combinatorial exercise which we leave for the reader. Lemma 7.3.2 enables one to compute the singular locus of the rings G , and, again by dehomogenization, that of R see 61], 6.B. We con ne ourselves to the rings Rr+1, for which there is a simpler approach. Suppose that x = (xij ) is an m n matrix over a ring R such that x11 is a unit. Then we may transform x by elementary row and column operations into the matrix 0 x11 0 0 1 BB 0 y11 y1n;1 CC y = x ; x x x;1 .. .. A ij i+1j +1 1j +1 i+11 11 @ ... . . 0 ym;11 ym;1n;1 and clearly Ir+1(x) = Ir (y). The equation yij = xi+1j +1 ; x1j +1xi+11x;111, read as a substitution of indeterminates, suggests the following elementary lemma. Lemma 7.3.3. Let X = (Xij ) and Y = (Yij ) be matrices of indeterminates over a ring B of sizes m n and (m ; 1) (n ; 1). Then the substitution Yij
7! Xi+1j+1 ; X1j+1Xi+11X11;1 yields an isomorphism ;1 ] B Y X11 . . . Xm1 X12 . . . X1n X11 = B X X11;1]
which for every t > 0 maps the extension of It;1(Y ) to the extension of It(X ). In particular it induces an isomorphism ;1] = Rt (X )x;1] (2) Rt;1(Y )X11 . . . Xm1 X12 . . . X1n X11 11 where x11 denotes the residue class of X11 in Rt(X ). Proposition 7.3.4. For a prime ideal Spec Rr+1 the localization is regular if and only if Ir (X )=Ir+1(X ). Proof. We use induction on r, starting from the trivial case r = 0 (note that I0 (X ) = kX ]). Suppose now that r > 0. If is the maximal ideal of R = Rr+1 generated by the xij , then R is evidently non-regular and contains Ir (X )=Ir+1(X ). Otherwise does not contain one of the residue classes xij , and by symmetry we may assume that x11 = . Then R is a localization of R x;111], and contracting the extension of via the isomorphism (2) to S = Rr (Y ) we obtain a prime ideal Spec S . As the extension from S to R x;111] is an adjunction of indeterminates followed
p
6
2
p
p
p
p
p
q
2 2
p
p
p
315
7.3. Properties of determinantal rings
by the inversion of one of them, R is regular if and only if S is regular. Furthermore Ir (X )=Ir+1(X ) if and only if Ir;1(Y )=Ir (Y ). The rings Rr+1 satisfy Serre's condition (R2) since height(Ir (X )=Ir+1(X )) = height Ir (X ) ; height Ir+1(X ) = m + n ; 2r + 1 3: By Serre's normality criterion this argument, together with the Cohen{ Macaulay property, proves independently of 7.3.2 that Rr+1 is a normal domain. (In fact, all the rings R and G satisfy (R2) see 61], (6.12).) The example m = n = 2, r = 1 shows that (R3) fails in general. Finally we want to determine which of the rings Rr+1 are Gorenstein. The easiest way to solve this problem is to determine the canonical module, or rather the divisor class it represents (see 3.3.19 the canonical module of Rr+1 is unique by (vi) below). In the following we use elementary facts from the theory of class groups of Noetherian normal domains R see 47], Ch. 7, or 108]. (i) The elements of Cl(R ) are the isomorphism classes I ] of fractionary divisorial ideals of R a fractionary ideal is divisorial if and only if it is a reexive R -module, and 2 Spec R is divisorial if and only if height = 1. One has I ] = 0 if and only if I is principal. In particular, R is factorial if and only if Cl(R ) = 0. (ii) The addition in Cl(R ) is given by I ] + J ] = (IJ )] where denotes the R -dual HomR ( R ). (iii) (`Gauss' lemma') The extension I ] 7! IR T ]] yields an isomorphism of class groups Cl(R ) = Cl(R T ]) (here T denotes an indeterminate over R ). (iv) (Nagata's theorem) If S R is multiplicatively closed, then the assignment I ] 7! IRS ] maps Cl(R ) surjectively onto Cl(RS ) the kernel of this map is generated by the classes ] of the divisorial prime ideals with \ S 6= . T (v) An ideal I R is divisorial if and only if I = ri=1 (i ei) with P divisorial prime ideals i one then has I ] = ri=1 ei i ] ( (e) is the e-th symbolic power R \ e R .) (vi) If R is a positively graded k-algebra with maximal ideal , then one has a natural isomorphism Cl(R ) = Cl(R ). It follows that the canonical module of R is unique (up to isomorphism) since this holds for R . Theorem 7.3.5. Suppose that 0 < r < min(m n), and let be the ideal of p
q
p
q
p
p
p
p
p
p
p
p
p
p
p
m
m
m
p
Rr+1 generated by the r-minors of the rst r rows of the residue class x of X, and the corresponding ideal for the rst r columns. Then (a) and are prime ideals of height 1, (b) Cl(Rr+1) is isomorphic to Z, and is generated by ] = ]. q
p
q
p
;
q
316
7. Determinantal rings
= R", " = 1 . . . r ;1 r +1 j 1 . . . r], together with 7.3.1 and the analogous isomorphism for Rr+1= . (b) Let = 1 . . . r j 1 . . . r], and = fxij : i r or j rg. We claim that is algebraically independent over k and that Rr+1 ;1] = k ;1]. In fact, one sees that xuv 2 k ;1 ] by expanding the minor 1 . . . r u j 1 . . . r v] along row u (or column v) in Rr+1 one has 1 . . . r u j 1 . . . r v] = 0. The algebraic independence of follows as in the ;1 Proof. (a) follows from the isomorphism Rr+1=
p
q
proof of 7.3.2. By (iii) and (iv) above, Cl(k ]) = 0 so that, again by (iv), Cl(Rr+1) is generated by the classes of those divisorial prime ideals which contain . The systems of generators of and speci ed in the theorem are ideals in the poset r+1 and their intersection is exactly f g. In conjunction with Exercise 7.1.8 this shows ( ) = \ . We conclude that ] = ; ] and that ] generates Cl(Rr+1). It remains to be shown that u ] 6= 0 for all u > 0. Suppose that u ] = 0, or, equivalently, that (u) is a principal ideal (a), a 2 R . Since (u) contains u, the extension (u)k ;1 ] equals k ;1]. Hence a is a unit in k ;1 ]. In k ] the element is the determinant of a matrix of indeterminates, and therefore a prime element according to 7.3.1. Thus a = e v with e 2 k and v 0. In the case where u > 0 we would have v > 0, and and would be minimal prime ideals of (u). It is now easy to reduce the computation of the canonical class of Rr+1 to the case r = 1 fortunately R2 is a normal semigroup ring, and we can draw on the results of Chapter 6. The hypothesis 0 < r < n m in the following theorem has been inserted in order to exclude the trivial cases r = 0 or r = min(m n). The condition n m is no restriction since we may replace X by its transpose if necessary. Theorem 7.3.6. With the notation of 7.3.5 suppose that 0 < r < n m. p
q
p
q
p
q
p
p
p
p
p
p
p
q
p
Then
(a) (m;n) is the canonical module of Rr+1, (b) (Svanes) Rr+1 is Gorenstein if and only if m = n. Proof. The canonical module of Rr+1 is uniquely determined as was observed above. In particular (a) implies (b). In proving (a) we rst suppose that r 2. The isomorphism (2) in 7.3.3, and (iii) and (iv) above induce a homomorphism Cl(Rr+1) ! Cl(Rr+1x;111]) = Cl(Rr (Y )X11 . . . Xm1 X12 . . . X1n X11;1]) = Cl(Rr (Y )) which maps the generator ] of Cl(Rr+1) to the analogous generator 0 ] of Cl(Rr (Y )) in particular it is an isomorphism. We set S = Rr+1x;111 ], and identify Rr+1 and Rr (Y ) with subrings of S . Let ! be the canonical module of Rr+1. Since the formation of the p
p
p
317
7.3. Properties of determinantal rings
canonical module commutes with localization, ! Rr S is a canonical module of S . Let !0 be a divisorial ideal of Rr (Y ) which under the above isomorphism has the same class as !. Then ! Rr S = ! 0 Rr ( Y ) S . As the extension Rr (Y ) ! S is faithfully at, 3.3.30 implies that !0 is a canonical module of Rr (Y ). Summing up, we conclude that u ] is the class of the canonical module of Rr+1 if and only if u 0 ] is the class of the canonical module of Rr (Y ). An iteration of the argument reduces (a) to the case r = 1. Suppose that r = 1. Let U1 . . . Um and V1 . . . Vn be indeterminates over k. The 2-minors of the matrix x = (Ui Vj ) vanish so that the substitution Xij 7! xij = Ui Vj induces a surjective homomorphism from R2 onto the normal semigroup ring kC ] generated by the monomials xij . An easy calculation of dim kC ] yields that we may in fact identify R2 and kC ]. Let I be the ideal generated by relint C in kC ]. By virtue of 6.3.5 I is the canonical module of kC ]. Let i be the prime ideal generated by the entries in the i-th row of x = (xij ), and j the corresponding ideal for the j -th column. Then the ideals i and j are exactly the height 1 ZC -graded prime ideals. In fact, the ZC -graded prime ideals are those prime ideals which are generated by some of the elements xij , and thus are the prime ideals generated by all the xij in the union of a set of columns and a set of rows. It follows from 6.1.1 and 6.1.6 (or direct arguments) that I = 1 \\ m \ 1 \ \ n . Therefore +1
+1
p
p
p
q
p
q
p
I ] =
m X
i] + p
i=1
p
q
q
Xn
j ] = m ] + n ] = (m ; n) ]: q
p
q
p
j =1
Remarks 7.3.7. (a) One can show that the symbolic and ordinary powers of the prime ideals and in 7.3.5 coincide so that m;n is the canonical module of Rr+1. (The case r = 1 is indicated in Exercise 7.3.9.) Furthermore 7.3.5 and 7.3.6 can be extended to all the rings R and G see 61]. (b) With the notation of 7.3.6 and its proof, I is the canonical module of R2 . But it is impossible to preserve the grading under the divisorial arguments by which we computed the canonical module of Rr+1 from that of R2 . In Bruns and Herzog 57] it has been shown that the a-invariant of Rr+1 is ;rm. As m;n is generated by elements of degree (m ; n)r, it follows that m;n (;rn) is the canonical module of Rr+1. (c) Let X be a symmetric n n matrix of indeterminates more precisely, the entries Xij of X with i j are algebraically independent, and Xij = Xji for i > j . The residue class rings Sr+1 = kX ]=Ir+1(X ) are as well understood as the rings Rr+1 constructed from `generic' m n matrices. In particular Sr+1 is a normal Cohen{Macaulay domain of dimension r(r + 1)=2 + (n ; r)r (Kutz 255]), its divisor class group is p
q
p
p
p
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7. Determinantal rings
Z=(2), and it is Gorenstein if and only if r + 1 n mod 2 (Goto 130],
131]) see Exercise 7.3.10. There is also a `standard monomial' approach to the structure of Sr+1, in which `doset algebras' replace ASLs (see De Concini, Eisenbud, and Procesi 73]). (d) Let X be a alternating n n matrix of indeterminates this of course means that the entries Xij of X with i < j are algebraically independent, Xii = 0, and Xij = ;Xji for i > j . The residue class ring Pr+2 of kX ] with respect to the ideal Pf r+2(X ), r even, is a normal Cohen{Macaulay domain of dimension r(r ; 1)=2 + (n ; r)r (Kleppe and Laksov 237]) it is factorial (Avramov 24]), and therefore Gorenstein by 3.3.19. The rings Pr+2 carry a `natural' ASL structure 73]. Exercises
Let R be a nitely generated faithfully at Z-algebra, and let P be one of the properties `Cohen{Macaulay', `Gorenstein', `reduced', `normal', `integral domain', (Sn ), (Rn ). (a) Show that the following are equivalent: (i) R k has P for every eld k (ii) R B has P for every Noetherian ring B which satis es P. (b) Suppose that K k L has P whenever K and L are extension elds of k one of which is nitely generated (for which of the listed properties is this true?). Show that (a)(ii) already follows from the fact that R has P. Hint: Exercise 5.1.25 is a similar problem. 7.3.9. With the notation of 7.3.5 assume r = 1. Show (i) = i for all i. Hint: 6.1.1. 7.3.10. With the notation of 7.3.7(c) let be the ideal generated by the r-minors of the rst r rows of X , and = 1 . . . r j 1 . . . r]. We use the fact that Sr+1 is a normal Cohen{Macaulay domain, and that is a prime ideal in Sr+1. Show (a) 2 ( ), (2) = ( ), and Cl(Sr+1) = Z=(2), (b) the canonical module of Sr+1 is Sr+1 if r + 1 n mod2, and otherwise, (c) Sr+1 is Gorenstein if and only if r + 1 n mod2. Hint: S2 can be considered as the second Veronese subring of a polynomial ring k Y1 . . . Yn ], or as a normal semigroup ring. 7.3.11. Let X be an m n matrix of indeterminates over a eld k with m n. Show that 1 . . . m] is a prime element in G(X ), and deduce G(X ) is factorial. 7.3.8.
p
p
p
p
p
p
p
Notes The notion of an algebra with straightening law is due to Eisenbud 90]. It was generalized to that of a (not necessarily graded) Hodge algebra in De Concini, Eisenbud, and Procesi 73]. This monograph contains all the theory developed in Section 7.1 as well as numerous examples of Hodge algebras. A signi cant class of non-ASL Hodge algebras are the coordinate rings of the varieties of complexes (De Concini and Strickland 75]). That the notion of a graded Hodge algebra is very general is illustrated by a theorem of Hibi 164]: every positively graded
Notes
319
ane algebra over a eld is a Hodge algebra. A non-graded Hodge algebra may behave pathologically as was shown by Trung 370]. The term `Hodge algebra' reects the fact that the rst standard monomial theory was created by Hodge 203] as a method for establishing the `postulation formula' for the Grassmannian and its Schubert subvarieties. In algebraic language this amounts to the computation of the Hilbert function of the rings G(X ) and G , and therefore is `only' a matter of counting the standard monomials of a xed degree. See also Hodge and Pedoe 204]. More recent accounts are due to Laksov 256] and Musili 282], whom we follow in proving the linear independence of the standard monomials. The straightening law on the polynomial ring is due to Doubilet, Rota, and Stein 77]. We follow De Concini, Eisenbud, and Procesi 72] in deriving it from that of G(X ). For a detailed account of the history of determinantal ideals we refer the reader to Bruns and Vetter 61], 2.E. It begins with Macaulay 262] who proved (in a special case) that the ideals Ir+1(X ) are unmixed for r + 1 = min(m n). In the inuential paper 86] Eagon and Northcott constructed a nite free resolution of these ideals and proved their perfection (which (over a eld of coecients) is equivalent to the Cohen{Macaulay property of the rings Rr+1 see 2.2.15). This resolution, the so-called Eagon{Northcott complex, has served as a model for several related constructions. In this connection one should mention the theory of generic perfection which was also developed by Eagon and Northcott 87] also see 61], Section 3. Its main result is that a `generic' acyclic complex remains acyclic under extensions of the ring of coecients. The Cohen{Macaulay property of the rings Rr+1 and their normality for general r are due to Hochster and Eagon 189]. They used an inductive scheme based on a `principal radical system'. That the rings G are Cohen{ Macaulay seems to have been realized independently by Hochster 177], Laksov 256], and Musili 282]. The Gorenstein determinantal rings were determined by Svanes 367] whereas the divisor class group and the canonical module were computed by Bruns 52], 55]. A driving force in the investigation of determinantal ideals was their relation to invariant theory: the rings Rr+1 and G(X ) appear as ring of invariants of `natural' linear group actions. In order to prove this fact (in arbitrary characteristic) De Concini and Procesi 74] established the straightening laws on which the ASL structures are built also see 61]. The Rees and associated graded rings of kX ] with respect to the ideals Ir+1(X ) are ASLs in a natural way and Cohen{Macaulay when r + 1 = min(m n) see Bruns, Simis, and Trung 60]. For r + 1 < min(m n) the Cohen{Macaulay property holds at least in characteristic zero, but fails in general 56]. The Hilbert function of Rr+1 and the numerical invariants derived from it are the subject of a monograph by Abhyankar 7]. See Herzog
320
7. Determinantal rings
and Trung 163] for an approach using Grobner bases. The homological properties of Rr+1 discussed in this chapter were proved by inductive methods. It would be much more satisfactory to derive them from a minimal free resolution of Rr+1 over kX ]. As pointed out above, in the case r + 1 = min(m n) the Eagon{Northcott complex is such a resolution, and for r + 1 = min(m n) ; 1 a suitable complex was constructed by Akin, Buchsbaum, and Weyman 8]. Both these complexes are characteristic-free they are de ned over Z and specialize to a minimal free resolution under base change from Z to an arbitrary eld. Recently Hashimoto 154] showed that such a resolution also exists for r + 1 = min(m n) ; 2. In characteristic zero Lascoux 257] described a minimal resolution of Rr+1 for arbitrary r. However, the construction of such resolutions seems to be exceedingly dicult in positive characteristics as is indicated by a result of Hashimoto 153]: for 1 < r +1 < min(m n);2 the Betti numbers of Rr+1 depend on the characteristic of k. The theory of determinantal rings has many aspects not considered in this chapter. For these, as well as for an extensive bibliography, we refer the reader to 61].
Part III
Characteristic p methods
321
8 Big Cohen{Macaulay modules
In this chapter we prove Hochster's theorem on the existence of big Cohen{Macaulay modules M for Noetherian local rings R containing a eld. An R -module is called a big Cohen{Macaulay module if there is a system of parameters x for which M is x-regular. Note that one does not require M to be nite, thus the attribute `big'. The importance of big Cohen{Macaulay modules stems from the fact that one can deduce many fundamental homological theorems from their existence (as we shall see in Chapter 9). Their construction is a paradigm for the application of characteristic p methods: one rst shows that big Cohen{Macaulay modules exist in characteristic p then the result is transferred to characteristic zero via a rather abstract principle. It asserts that certain `generic' systems of equations are soluble over some local ring of characteristic p if there is a solution in characteristic zero. Rings of characteristic p are endowed with a canonical endomorphism, the Frobenius homomorphism a 7! ap . Its homological power seems to have rst been realized by Peskine and Szpiro. They also introduced M. Artin's approximation theorem to commutative algebra. The approximation theorem guarantees the descent from complete, `analytic' local rings to `algebraic' ones. 8.1 The annihilators of local cohomology Let (R ) be a Noetherian local ring and let i i = AnnR H (R ) be the annihilator of the i-th local cohomology. This notation is kept throughout the section. As we shall see, the products 0 j annihilate the homology of certain complexes, and furthermore 0 n;1 annihilates the ideals (x1 . . . xj ): xj +1 modulo (x1 . . . xj ) for all systems x1 . . . xn of parameters and j = 0 . . . n ; 1. This will be important in the construction of big Cohen{Macaulay modules in characteristic p. Theorem 8.1.1. Let R be a Noetherian local ring of dimension n which is a residue class ring of a Gorenstein local ring S, dim S = d. Then the following hold for i = 0 . . . n: m
a
m
a
a
a
323
a
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8. Big Cohen{Macaulay modules
(a) i = AnnR ExtdS ;i(R S ) (b) dim R= i i (c) 0 n;1 contains a non-nilpotent element (d) for 2 Spec R, dim R= = i, one has 2 Ass R= i () 2 Ass R. Proof. (a) We want to show rst that both R and S can be replaced by their completions R^ and S^ for the proof of (a). Of course one has R^ = R S S^ , and the formation of local cohomology commutes with completion: by 3.5.4, H i (R ) = H i^ (R^ ): = H i (R ) R R^ a
a
a
a
p
p
p
a
m
m
p
m
The same holds for Ext since S^ is a at S -module and R has a resolution by nite free S -modules. Finally, for every R -module N , one has AnnR N = R \ (AnnR^ (N R R^ )) because R^ is faithfully at. So we may assume that S and R are complete. We saw in the proof of 3.5.7 that H i (R ) = H i (R ) for all i (as an S or R -module), denoting the maximal ideal of S . Let E be the injective hull of S= over S , and 0 the functor HomS ( E ). Since H i (R ) = H i (R )00 we have m
n
n
n
n
n
AnnR H i (R ) AnnR H i (R )0 AnnR H i (R )00 = AnnR H i (R ) n
n
n
n
and so the local duality theorem 3.5.8, applied to the S -module R , yields AnnR H i (R ) = AnnR H i (R )0 = AnnR ExtdS ;i (R S ): n
n
(b) This inequality is 3.5.11(c). (c) By (b) one has dim(R=( 0 n;1)) n ; 1. Therefore 0 n;1 is not contained in any minimal prime ideal with dim R= = n. (d) Consider the preimage of in S . Since dim S = d ; i, one has depth R = depth R = 0 if and only if ExtdS ;i (R S ) = ExtdS ;i (R S ) 6= 0. Consequently 2 Ass R if and only if 2 Supp ExtdS ;i (R S ). Because of (a) and (b) the latter is equivalent to 2 Ass R= i for prime ideals such that dim R= = i. A very important property of the ideals i is expressed by the following theorem. Theorem 8.1.2. Let R be a Noetherian local ring, and a
a
a
p
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a
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q
p
q
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a
F. : 0
;! Fm ;! Fm;1 ;! ;! F0 ;! 0
a complex of nitely generated free R-modules such that all the homology modules Hi (F. ) have nite length. Then 0 m;i annihilates Hi (F. ) for i = 0 . . . m. a
a
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8.1. The annihilators of local cohomology
Proof. First we construct an object which connects the local cohomology of R and the homology of F. . Let x1 . . . xn be a system of parameters,
and let K denote the complex 0 ;! Kn ;! ;! K1 ;! K0 ;! 0 M Kj = Rxi xin;j : .
1
1 i1 < m or t ; p > n. where U p =U p;1 = Ept ;p pt;p Thus the ltration is already given by 0 = U t;n;1 U t;n U m = Ht (T ) .
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and Ht (T ) is annihilated by n;(t;(t;n)) n;(t;m) = 0 n;t+m. Taking into account that Hi (F ) = Hi;n (T ) we get the desired result. As a consequence we derive another `annihilation theorem' whose second part is crucial in the construction of big Cohen{Macaulay modules in characteristic p. Corollary 8.1.3. Let R be a Noetherian local ring of dimension n. Then, given a sequence x = x1 . . . xm 2 R such that codim(x1 . . . xm ) = m, the .
a
a
.
a
a
.
following hold: (a) 0 n;i annihilates the Koszul homology Hi (x), i = 0 . . . m (b) 0 n;1 annihilates ((x1 . . . xm;1) : xm )=(x1 . . . xm;1). a
a
Proof.
a
a
(a) The sequence x can be extended to a system of parameters
;
x1 . . . xn (recall that codim I = dim R dim R=I ). We start a descending induction at m = n for which the assertion is obviously a special case of
8.1.2. Suppose now that m < n and put x0 = x1 . . . xm xtm+1, t 1.6.13 we have an exact sequence
1.
By
x ' ;;;! Hi (x) ;! Hi (x0): By induction the submodule Im ' = Hi (x)=xtm+1Hi (x) of Hi (x0 ) is annihiT t lated by 0 n;i. Since xm+1Hi (x) = 0, we are done.
Hi (x)
t m+1
(b) We use another segment of the long exact sequence of Koszul homology, now relating x and x00 = x1 . . . xm;1 : a
a
x ;! H0 (x00 ) ;! H0 (x00 ): 0 n;1H1 (x) = 0, we also have 0 n;1(Im ) = 0. 00
H1 (x)
m
Since Im consists of exactly those elements in H0 (x ) = R=(x1 . . . xm;1) annihilated by xm , that is Im = ((x1 . . . xm;1) : xm )=(x1 . . . xm;1). The preceding corollary is completely vacuous if R happens to be a Cohen{Macaulay ring, but in connection with 8.1.2 it shows that certain local rings, among them the complete ones, preserve a faint trace of the Cohen{Macaulay property: the modules ((x1 . . . xj ;1) : xj )=(x1 . . . xj ;1) which are zero for R Cohen{Macaulay, cannot be arbitrarily `big'. Corollary 8.1.4. Let R be a Noetherian local ring which is a residue class a
a
a
a
ring of a Gorenstein local ring S. Then there exists a non-nilpotent element c R such that c ((x1 . . . xj ) : xj +1)=(x1 . . . xj ) = 0 for all systems of parameters x1 . . . xn and all j = 0 . . . n 1. Proof. According to 8.1.1 there exists a non-nilpotent c 0 n;1. By 8.1.3 such a c satis es our needs.
2
;
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8.2. The Frobenius functor
Remark 8.1.5. Parts (b) of 8.1.1 and 8.1.4 are not true for arbitrary Noetherian local rings. One of the most used counterexamples of commutative algebra (constructed by Nagata 284], Example 2, p. 203) works here, too: let R be a 2-dimensional local domain such that its completion R^ ^ = 1. Put 1 = AnnR H 1 (R ) has an associated prime ideal with dim R= and 1 = AnnR^ H 1^ (R^ ). Since H 1 (R ) = H 1^ (R^ ) as R -modules, 1 = 1 \ R . ^ 1 . So Of course 8.1.1 applies to R^ , and by its third part, 2 Ass R= = 2. (Note that a regular element of R \ R = 0 and dim R= 1 1 ^ stays regular in R .) Let x = x y be a system of parameters of R and put xt = xt yt . Then H 1 (R ) is the direct limit of the Koszul cohomology modules H 1 (xt ). By 1.6.10 one has H 1 (xt) = H1 (xt). Consider the long exact sequence (which appeared already in the proof of 8.1.3): p
p
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H1 (yt )
x ;! H1(xt) ;! H0(yt) ;! H0 (yt ): t
R is a domain, so H1 (yt ) = 0, and H1 (xt ) = (yt : xt )=(yt). An element c annihilating all the modules (yt : xt )=(yt) must annihilate H 1 (R ), hence c = 0 as seen above. m
Exercises Let M be an arbitrary module over a Noetherian local ring (R ), and (M ) = Ann H i (M ). Let F. be a complex of nite free R -modules with homology of nite length as in 8.1.2. Prove that 0 (M ) m i (M ) annihilates Hi (F. M ) for i = 0 . . . m. 8.1.7. With R and M as in 8.1.6 assume that H i (M ) = 0 for i = 0 . . . n ; 1 where n = dim R , and M= M 6= 0. Let x be a system of parameters of R . Show Hi (x M ) = 0 for i = 1 . . . n, and that x is M -quasi-regular. (See 1.6.20.) 8.1.6. a
i
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;
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8.2 The Frobenius functor Let R be a ring of characteristic p, i.e. a ring with a monomorphism Z=pZ ;! R where p is a prime number. The Frobenius homomorphism is the map F : R ;! R , F (a) = ap. Via F one may consider R as an R algebra in a non-trivial way. The crucial point in the construction of the Frobenius functor is to work simultaneously with two essentially di erent module structures of R itself. This is an unusual idea in commutative algebra and has to be kept rmly in mind. Let R F denote the (R ;R )bimodule with additive group R and left and right scalar multiplication given by a r b = arF (b) = arbp a b 2 R r 2 R F : (The standard associative laws are obviously satis ed.)
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Let M be a left R -module. Then we take the tensor product R F R M with R F as a right R -module, i.e. a bx = a b x = abp x a 2 R F b 2 R x 2 M: The left R -module structure of R F endows R F R M with a like structure such that c(a x) = ca x. (This tensor product is merely biadditive bilinearity is lost because in general a r 6= r a for a 2 R , r 2 R F .) The Frobenius functor F acts on a left R -module M by assigning to it the left R -module R F R M . For an R -linear map ' : M ! N one consequently considers F(') to be the R -linear map idRF R '. The following properties of F are just the fundamental ones of tensor products. Proposition 8.2.1. Let R be a ring of characteristic p. Then F is a covariant,
additive, and right exact functor from the category of left R-modules to itself.
We want to compute some speci c values of F. First we see that F(R) = RF nR R = nRF as a left R-module, so F(R) = R then additivity implies F(R ) = R . For a cyclic R -module R=I one gets F(R=I ) = R F R R=I = R F =(R F I ). Now r a = rap for r 2 R F , a 2 I , and R F I turns out to be the ideal I p] generated by the p-th powers of the elements of I . Hence R F =(R F I ) = R=I p] with its ordinary left scalar multiplication. Proposition 8.2.2. Let R be a ring of characteristic p. Then (a) F(R n) = R n for all n (as left R-modules), and if e1 . . . en is a basis of R n, 1 e1 . . . 1 en is a basis of F(R n), (b) F(R=I ) = R=I p] for all ideals I of R. More generally, we denote by I q] , q = pe , the ideal generated by the q-th power of the elements of I I q] is called the q-th Frobenius power of I. The Frobenius functor owes its power to its non-linearity, again something remarkable. It is straightforward to verify the following. Proposition 8.2.3. Let R be a ring of characteristic p, M and N be Rmodules, and ' : M ! N an R-linear map. Then (a) F(a') = apF(') for all a 2 R, P (b) P apif(1'(xy) )=, aiyi for x 2 M, ai 2 R yi 2 N, then F(')(1 x) = i i (c) the map M ! F(M ), x 7! 1 x, is not R-linear in general: instead one has (ax) 7! ap (1 x). We can now give a concrete description of F in terms of `generators and relations':
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Proposition 8.2.4. Let R be a ring of characteristic p and M an R-module ' with a presentation R m ;! R n ;! M ;! 0. F(') (a) Then F(M ) has the presentation R m ;;! R n ;! F(M ) ;! 0 (b) furthermore, if ' is given by a matrix (aij ), then F(') is given by the matrix (apij ). Part (a) follows from the right exactness of F and the fact that F leaves R n untouched. Part (b) follows from 8.2.3. We conclude the list of basic properties of the Frobenius functor with its behaviour under localization: Proposition 8.2.5. Let R be a ring of characteristic p. The Frobenius functor commutes with rings of fractions: RS R F(M ) = F(RS R M ) for all Rmodules M, and analogously for R-linear maps. Proof. We have RS R (M ) = RS R R F R M and
F F(RS R M) = RSF R RS R M = RSF R M: As left RS -modules, RS R R F = RS R R = RS and RSF are naturally S
isomorphic this isomorphism is also an isomorphism of right R -modules.
We cannot resist trying the strength of the Frobenius functor by proving the `new intersection theorem' in characteristic p. (This nomenclature will be explained in Chapter 9.) The elegant argument, including 8.1.2, is due to Roberts. Theorem 8.2.6. Let (R ) be a Noetherian local ring of characteristic p, m
and
' ;! ;!
' ;! ;! ;! ;!
F. : 0 Fs Fs;1 F1 F0 0 a complex of nite free R-modules such that each homology Hi (F. ) has nite length. If s < dim R, the complex F. is exact. Proof. Note that it is enough to cover the case of a complete ring R : if R is not complete, we simply tensor all our objects by R^ , a faithfully at s
1
extension of the same dimension. Assume that F is not exact. If H0 (F ) = 0, the map '1 is a split epimorphism, and F decomposes into the direct sum of two shorter complexes of the same type. So we may suppose H0 (F ) 6= 0. Furthermore, if '1 (F1) 6 F0, F splits o an isomorphism '01 : F10 ! F00 of direct summands of F1 and F0. This leaves the essential case '1(F1 ) F0. Apply the Frobenius functor F to F . The modules appearing in F(F ) are the same as in F . Furthermore F(F ) has nite length homology: by hypothesis F R is split exact for all prime ideals 2 Spec R , 6= . Since the Frobenius functor commutes with localization, .
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this also holds for F(F ). Something has changed however, namely we have F('1)(F1) p F0 by 8.2.4(b). Now one iterates this procedure: all the complexes Fe(F ) e 0, e e p have nite length homology, and F ('1)(F1) F0. On the other hand H0 (Fe(F )) is annihilated by the ideal 0 s T wheree i = Ann H i (R ) p = 0, contradicting see 8.1.2. This forces 0 s to be contained in 8.1.1 for s < dim R since R , a complete local ring, is a residue class ring of a Gorenstein ring. A crucial point in the preceding proof is that for a nite free complex F the Frobenius functor F preserves the property of having nite length homology. It also preserves acyclicity: Theorem 8.2.7 (Peskine{Szpiro). Let R be a Noetherian ring of character's ' istic p, and F : 0 ;! Fs ;! ;! F0 a complex of nite free R-modules. Then F is acyclic if and only if F(F ) is acyclic. P Proof. Set rj = si=j (;1)i;j rank Fi . By the acyclicity criterion 1.4.13 it depends only on the grades of the ideals Iri ('i) whether F is acyclic or not: it is acyclic if and only if grade Iri ('i) i for i = 1 . . . s. By virtue of 8.2.2 and 8.2.4, rst F(Fj ) = Fj , and next Iri (F('i)) = Iri ('i )p]. The two ideals have the same radical, hence the same grade. .
m
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The following corollary will play an important r^ole in Chapter 10. Corollary 8.2.8 (Kunz). Let R be a regular ring of characteristic p. Then
F is an exact functor. Proof. Since atness is a local property and F commutes with localizaR F is a at R-algebra equivalently,
tion, we may assume that R is a regular local ring. By a standard atness criterion (for example, see 270], 7.8) it is enough that TorR1 (R F R=I ) = 0 for every ideal I of R . This follows from 8.2.7: the nite free resolution of R=I stays acyclic when tensored with R F . The assertion of 8.2.8 is usually called the atness of the Frobenius. Kunz 245] also showed the converse of 8.2.8 see Exercise 8.2.11 for the case in which R is Cohen{Macaulay. Exercises Let R be a Noetherian ring, M a nite R -module of nite projective dimension with nite free resolution F. , and e 1 an integer. Prove that Fe (F. ) is acyclic, and proj dim Fe (M ) = proj dim M . 8.2.10. Let R be a regular local ring of characteristic p. Show `(Fe (M )) = pe dim R `(M ) for every nite length module M . (Use induction on `(M ).) 8.2.9.
8.3. Modi cations and non-degeneracy
331
Herzog 157] proved that 8.2.9 characterizes modules of nite projective dimension then it follows immediately from 2.2.7 that the exactness of F characterizes the regular ones among the Noetherian local rings. For simplicity we restrict ourselves to Cohen{Macaulay rings. So suppose R is a Cohen{ Macaulay local ring of characteristic p. (a) Let F. be a minimal free resolution of a nite R -module M and x a maximal R -sequence. If Fe (F. ) is acyclic for all e 1, show TorRi (R=(x) Fe (M ))
= (R=(x))bi (M) for i > 0 and e 0. (b) Conclude that proj dim M < 1.
8.2.11.
8.3 Modi cations and non-degeneracy In this section we show that for a system of parameters x = x1 . . . xn of a Noetherian local ring of characteristic p there exists an x-regular R -module. The conditions to satisfy are: (i) xs+1 is a regular element of M=(x1 . . . xs )M , s = 0 . . . n ; 1, (ii) M 6= xM . Since the trivial choice M = 0 satis es (i), we see that (i) is completely useless without (ii), and we need results from the preceding sections in order to show that the construction below does not degenerate by violating condition (ii). Suppose M is an R -module such that xs+1 is not (M=(x1 . . . xs )M )regular. Then there exists a y 2 M y 62 (x1 . . . xs )M , for which xs+1y 2 (x1 . . . xs )M . Equivalent to y 62 (x1 . . . xs )M is the non-existence of a solution z1 . . . zs 2 M of the equation y = x1 z1 + + xs zs . The deus ex machina by which algebraists force equations to be soluble is to extend the given object by some `free' variables and to introduce the as yet insoluble equation as a relation on them. In our case we pass to M 0 = (M R s )=Rw, where w = y ; (x1 e1 + + xs es ), and e1 . . . es a basis of R s . The element y0 , the image of y under the natural map M ! M 0 , no longer keeps xs+1 from being regular on M 0 =(x1 . . . xs )M 0 . It is quite obvious that a well organized iteration of this construction in the limit f satisfying condition (i) of x-regularity. yields a module M f. It is however equally obvious that we may lose condition (ii) for M One attempt to control (ii), successful in characteristic p, is to keep track of a xed element f 2 M on its way to the limit and to make sure that f 62 xMi for all approximations Mi. For a pair (M f ) f 2 M , let M 0 be constructed as above, and f 0 be the image of f under the natural map M ! M 0 . Then (M 0 f 0) is called an x-modi cation of (M f ) (of type s). More generally, if there is a sequence (M f ) = (M0 f0 ) ;! (M1 f1) ;! ;! (Mr fr ) = (N g) with (Mi+1 fi+1) an x-modi cation of (Mi fi) (of type si+1), then (N g) is an x-modi cation of (M f ) (of type (s1 sr )). As soon as x is xed, we may simply speak of a modi cation. If g 62 xN , then (N g) is nondegenerate.
332
8. Big Cohen{Macaulay modules
Proposition 8.3.1. Let R be a Noetherian ring, and x = x1 . . . xn Then the following are equivalent: (a) there exists an x-regular R-module M (b) every x-modi cation (N g) of (R 1) is non-degenerate.
2 R.
We start with the more important implication (b) ) (a). Our goal is to construct a direct system of modules (Mi 'ij ), i 2 N, starting from M0 = R such that M = lim ;! Mi is x-regular. Each (Mi '0i(1)) will be a modi cation of (R 1). Therefore our hypothesis (b) forces lim '0i(1) 62 xM . ;! Suppose that M1 . . . Mj (together with the natural maps in a sequence of modi cations) have been determined. Now choose rst i, then s, minimal such that there exists a y 2 Mi with xs+1'ij (y) 2 (x1 . . . xs )Mj while 'ij (y) 62 (x1 . . . xs )Mj . Then put Mj +1 = (Mj R s)=Rw, w = y ; (x1e1 + + xs es ) as above, the maps 'ij +1 being the natural ones. Let us say that step j + 1 has index (i s). We claim that for each pair (i s) there are only nitely many steps of index (i s). For, if the sequence j + 1 . . . of steps of index (i s) does not stop, one nds a non-stationary ascending chain of submodules of Mi by taking the preimages of (x1 . . . xs )Mj +1 (x1 . . . xs )Mj +2 . . . in Mi . But R is Noetherian, and all the modules Mi are nite. If there is an equation xs+1y = x1z1 + + xs zs for elements y z1 . . . zs of the limit M , it has to hold in an approximation Mi as well. According to the claim, 'ij (y) 2 (x1 . . . xs )Mj for j i, hence y 2 (x1 . . . xs )M . The validity of the implication (a) ) (b) is forced by our choice of a free direct summand in the construction of a modi cation. Let f 2 M be any element 62 xM . Trivially there is a homomorphism R ! M , 1 7! f . So it is enough to show that if (N 0 g0 ) is a type s modi cation of (N g) and there is a map ' : N ! M , '(g) = f , then this map can be extended to '0 : N 0 ! M , '0 (g0 ) = f . Suppose that N 0 = (N R s )=Rw, w = y ; (x1e1 + + xs es ). Since xs+1'(y) 2 (x1 . . . xs )M and M is xregular, there are elements e01 . . . e0s 2 M such that '(y) = x1 e01 + + xs e0s . Thus take '0 to be the map induced by ' and the assignment ei 7! e0i . Proof.
Of course, the implication (a) ) (b) of the preceding proposition does not help in the construction of a big Cohen{Macaulay module. However, the idea in its proof, namely to compare a sequence of modi cations to some `universal' object, is very useful: Lemma 8.3.2. Let R be a Noetherian local ring which is a residue class
2
ring of a Gorenstein ring. Then there exists a non-nilpotent element c R such that for every system of parameters x and every sequence (R 1) = (M0 f0) (Mr fr ) of x-modi cations one has a commutative dia-
! !
333
8.3. Modi cations and non-degeneracy gram (M0 f0)
?? ;;! (M1?? f1) ;;! ;;! (Mr;1?? fr;1) ;;! (M??r fr ) y' y' y' ; y' 0
(R c)
c c ;;! ;;!
r
r
1
(R cr;1) ;;! (R cr ) the commutativity including 'i (fi) = ci , i = 0 . . . r. Proof. Take c as in 8.1.4, i.e. non-nilpotent and c ((x1 . . . xs ) : xs+1)=(x1 . . . xs ) = 0 for every system of parameters x and s = 0 . . . dim R ; 1. Naturally '0 = id. Suppose 'i has been chosen. If Mi+1 = (Mi R s )=Rw w = y ; (x1e1 + + xs es ) xs+1y = x1z1 + + xs zs zj 2 Mi then 'i (y) 2 (x1 . . . xs ) : xs+1, and there are elements e01 . . . e0s 2 R for which c'i (y) = x1 e01 + + xs e0s . The homomorphism '0 : Mi R s ! R , '0(g) = c'i (g) for g 2 Mi , '0 (ei ) = ei0 , factors through Mi+1, yielding the desired map 'i+1. Suppose R has characteristic p, and let F denote the Frobenius functor. Given an R -module M and f 2 M , we write F(f ) for 1 f 2 F(M) = RF M. We want to investigate how modi cations behave under F. With the standard meanings of y zi w M 0 we have xps+1F(y) = xp1 F(z1) + + xps F(zs) F(w) = F(y) ; (xp1F(e1) + + xpsF(es)) F(M0) = (F(M) F(Rs))=RF(w) and F(e1) . . . F(es ) form a basis of F(R s) = R s see 8.2.1 { 8.2.3. This 0 0 shows that if (M f ) is an x-modi cation of (M f ), then (F(M 0) F(f 0)) is an xp -modi cation of (F(M ) F(f )). All the arguments necessary to prove the existence of big Cohen{ Macaulay modules in characteristic p have now been collected. Let R be a Noetherian local ring of characteristic p. Note that a system of parameters x of R is a system of parameters of R^ , and every x-regular R^ module is also an x-regular R -module. Therefore we may assume that R is complete. According to 8.3.1 the existence of a degenerate x-modi cation (N g) of (R 1) must be excluded. Suppose (N g) is degenerate and of type (s1 . . . sr ). Let us now iterate the application of the Frobeniuse functor to the given data. After the e-th iteration we have reached an xp modi cation (N (e) g(e) ) of (R 1). Since (N g) is degenerate, i.e. g 2 xN , (N (e) g(e) ) is degenerate, too, i.e. g(e) 2 xpe N (e) . Since R is complete, 8.3.2 (R 1)
c ;;!
1
c
334
8. Big Cohen{Macaulay modules
can be invoked: there exists a homomorphism 'r : Ne (e) ! R such that e (e) r ( e ) ( e ) p r cT = e 'r (g ). However, g 2 x N , so c 2 (xp ) for all e. Since (xp ) = 0, c must be nilpotent { a contradiction. Theorem 8.3.3. Let R be a Noetherian local ring of characteristic p > 0, and x a system of parameters. Then there exists an x-regular module M. In particular R has a big Cohen{Macaulay module.
An equational criterion for degeneracy of modi cations. In the coming
section we want to derive the existence of big Cohen{Macaulay modules in characteristic zero from their existence in characteristic p. The key argument will be Hochster's niteness theorem which guarantees the solubility of certain systems of polynomial equations over some local ring of characteristic p provided there is a solution in characteristic zero. The following proposition gives a suciently detailed description of the equations to be used. Combined with 8.3.1 it is a criterion for the existence of x-regular modules, in particular the existence of big Cohen{Macaulay modules. Proposition 8.3.4. Let n 2 Z, n 1, and let s1 . . . sr 2 Z with 0 s1 . . . sr n ; 1. Then there exists a set S(s1 . . . sr ) of polynomials p 2 ZX1 . . . Xn Y1 . . . Ym ], m determined by s1 . . . sr , such that for every ring R and every sequence x = x1 . . . xn 2 R the following are equivalent: (a) there is a degenerate x-modi cation (N g) of (R 1) of type (s1 . . . sr ) (b) there exist y1 . . . ym 2 R such that p(x1 . . . xn y1 . . . ym ) = 0 for all p 2 S(s1 . . . sr ). Proof. Consider a sequence (R 1) = (M0 f0) ;! ;! (Mr fr ) = (N g) (Mi fi ) being a modi cation of (Mi;1 fi;1) of type si . Mi is constructed by adding generators ei1 . . . eisi and a relation wi = yi ; (x1 ei1 + + xsi eisi ) to Mi;1, so r M Xr N = Mr = ( Fj )= Rwv : j =0
v=1
The module M0 = R is simply generated by e10 = 1, and Fi has the basis ei1 . . . eisi . Writing yi as a linear combination of the basis elements one obtains alij 2 R such that
9> >> yi = = j =0 l=1 s i;1 X > X wi = ajil ejl ; (x1ei1 + + xs eis ) > > j =0 l=1 s i;1 X X j
(1)
ajil ejl
j
i
i
i = 1 . . . r:
335
8.3. Modi cations and non-degeneracy
The condition xsi +1yi 2 (x1 . . . xsi )Mi;1 can be formulated in xsi +1yi =
Xs i
u=1
xu giu +
i;1 X v=1
Li;1 Fj : j =0
bvi wv
L with giu 2 ij;=01 Fj , bvi 2 R . Expressing the giu in the given basis of Li;1 F and substituting the right sides of (1) for y and w yields j =0 i;1
i
s i;1 X X j
(2) =
i
xsi +1ajil ejl
j =0 l=1 si i;1 sj
XXX u=1 j =0 l=1
j xu cuj il el +
s i;1 X v ;1 X X j
v=1 j =0 l=1
bvi ajvl ejl
; X bvi xl evl sv
l=1
i = 1 . . . r:
L
Each of these equations relating elements of the free module ij;=01 Fi splits into its components with respect to the elements of the given basis. Replacing the coecients ajil bvi cujil xu by algebraically independent elements Ajil Biv Ciluj Xu over the ring Z and collecting all the terms in the components of (2) on one side, one obtains a set S0(s1 . . . sr ) of polynomials over Z which depends only on (s1 . . . sr ). We have seen that an x-modi cation of type (s1 . . . sr ) leads to a solution of the system S0(s1 . . . sr ) in which the variables Ajil Biv Ciluj take values in R whereas x1 . . . xn are substituted for X1 . . . Xn. Conversely, given such a solution, one de nes the elements yi and wi by their representations in (1). The validity of (2) then guarantees that one has constructed a chain of modi cations. Next we write down what it means for (Mr fr ) to be degenerate. The element fr is just the residue class of e10 = 1 in Mr . Therefore (Mr fr ) is degenerate if and only if e10
2 x(M Fj ) + X Rwv : r
r
j =0
v=1
This adds an (r + 1)-th relation to the system (2): e10 =
Xn Xr Xs j
u=1 j =0 l=1
xu cruj+1lejl +
s v;1 X Xr X
(
j
v=1 j =0 l=1
bvr+1ajvl ejl
; X bvr+1xl evl ): sv
l=1
Accordingly we enlarge our sets of indeterminates and of equations. In view of what must be proved, there is no need to distinguish the variables Ajil Biv Ciluj . We order them in some sequence and rename them Y1 . . . Ym .
336
8. Big Cohen{Macaulay modules
Exercise Let (R k) be a Noetherian local ring of dimension n, M a big Cohen{ Macaulay module, and y a system of parameters. Prove (a) H i (M ) = 0 for i = 0 . . . n ; 1 and H n (M ) 6= 0, (b) Hi (y M ) = 0 for i = 1 . . . n and H0 (y M ) 6= 0, (c) y is M -quasi-regular, (d) ExtiR (k M ) = 0 for i = 0 . . . n ; 1 and Extnk (k M ) 6= 0. (Use 8.1.7.) 8.3.5.
m
m
m
8.4 Hochster's niteness theorem The following theorem is fundamental for the application of characteristic p methods to (local) rings containing a eld of characteristic zero. Let X = X1 . . . Xn, Y = Y1 . . . Ym be families of independent indeterminates over Z. Then we call a subset E of ZX Y ] simply a system of equations (over Z). It has a solution of height n in a Noetherian ring R if there are families x = x1 . . . xn, y = y1 . . . ym in R such that (i) p(x y ) = 0 for all p 2 E
(ii) height xR = n:
Note that condition (ii) implies xR 6= R (by de nition, height R = 1). Theorem 8.4.1 (Hochster). (a) Suppose that the system E of equations has
E
a solution of height n in a Noetherian ring R containing a eld. Then has a solution x0 y 0 in a local ring R 0 of characteristic p > 0 such that x0 is a system of parameters for R 0. Moreover, R 0 can be chosen as a localization of an ane domain over a nite eld with respect to a maximal ideal. (b) If, in addition, R is a regular local ring such that x is a regular system of parameters, then the ring R 0 in (a) can be chosen as a regular local ring with regular system of parameters x0 .
The theorem suggests the following strategy for proving a statement
S about Noetherian rings containing a eld: (i) prove S for local rings of characteristic p > 0, (ii) show that there exists a family (Ei)i2I of systems of equations with the property that S holds for R if and only if none of the systems Ei has a solution of the appropriate height in R . Both steps (i) and (ii) have been carried out for the statement `If R is local and x a system of parameters, then there exists an x-regular R -module' see 8.3.1, 8.3.4, and 8.3.3. Thus one obtains Theorem 8.4.2 (Hochster). Let R be a Noetherian local ring containing a
eld, and x a system of parameters for R. Then there exists an x-regular R-module M. In particular, R has a big Cohen{Macaulay module.
337
8.4. Hochster's niteness theorem
The proof of 8.4.1 falls into three parts: (i) the reduction to its part (b), (ii) the reduction to the case in which R is the localization of an ane algebra, and (iii) the nal step. Before we set out for the proof of 8.4.1, we show that certain conditions of linear algebra over a regular local ring can be formulated by stating that the ring elements involved and some auxiliary elements satisfy a suitable system of equations. The equational presentation of acyclicity. Some conditions for elements in a ring R and vectors and matrices formed by them are evidently of an
equational nature: for example, the membership of a vector in a nite submodule of a free module of nite rank (especially that of an element in a nite ideal) and the assertion that a sequence of matrices forms a complex. The crucial fact for regular local rings is that the acyclicity of a complex can also be cast into equational conditions: Lemma 8.4.3. Suppose R is a regular local ring with regular system of parameters x1 . . . xn , and let the matrices 'p . . . '1 represent the linear
maps in a nite free resolution over R. Denote the family of the entries of (i)). all the 'i by z = (zjk Then there is a set of polynomials over Z in the indeterminates X = X1 . . . Xn, the family of indeterminates Z = (Zjk(i)) representing the entries of all the matrices 'i , and auxiliary indeterminates W = W1 . . . Wu such that the following holds: (a) there are w = w1 . . . wu R for which x z w is a solution of (b) whenever x0 z 0 w0 are specializations of X Z W in a regular local ring R 0 satisfying the systems of equations and such that x0 is a regular system (i)0 of parameters, then the complex formed by the matrices '0i with entries zjk is acyclic.
A
2
A
A
Proof.
We use the Buchsbaum{Eisenbud acyclicity criterion 1.4.13. Let
ri be the (expected) rank of 'i . Then grade Iri ('i ) i. For i dim R this is equivalent to the existence of elements aij (x), j = 1 . . . n i, for which Iri ('i) + (ai1 . . . ain;i) contains a power of each of the xl . For i > dim R , the condition grade Iri ('i ) i just says 1 Iri ('i). Thus the
2
2
;
grade condition of the acyclicity criterion can be interpreted equationally. In R 0 we use the converse direction of the acyclicity criterion. This lemma describes very precisely which indeterminates and equations between them must be introduced in order to transfer objects of linear algebra represented by matrices and some of their properties P from the regular local ring R to another regular local ring R 0 of the same dimension via a generic presentation as in 8.4.3. Let us simply say that P has a regular equational presentation. The following corollary lists some properties with regular equational presentations.
338
8. Big Cohen{Macaulay modules
Corollary 8.4.4. Let (R ) be a regular local ring, U V submodules of R r , v 2 R r , W a submodule of R s , and : R r ! R t a linear map. Then the m
following properties have regular equational presentations: (a) U = Ker (b) V =U = R s =W (c) v = U (d) ExtiR (R r =U R ) = R s =W for some xed i (e) dim R r =U = d for some integer d. Proof. (a) We choose a system u1 . . . uq of generators of U and de ne the linear map : R q R r by sending the i-th basis vector to ui. Then the sequence R q Rr R t can be extended to a nite free resolution F. of Coker . By virtue of the previous lemma the acyclicity of F. has a
2
! ;! ;!
regular equational presentation, and the acyclicity includes the condition U = Ker . (b) The given isomorphism V =U = R s=W and the choice of a system of generators of U as above induce a commutative diagram 0
;;;;! R?q ;;;;! R?q+s ;;;;! R s ;;;;! 0 ?? ? ?
y
y
y
0 ;;;;! U ;;;;! V ;;;;! V =U ;;;;! 0 with exact rows and epimorphisms . Conversely, given such a diagram, one has Ker = (Ker ). In other words, a system of generators of W is obtained by projecting a system of generators of Ker onto the last s coordinates. Let T = Ker . After the speci cation of matrices for it is sucient that the condition T = Ker has a regular equational presentation, and this is warranted by (a) since can also be considered as a linear map with target R r . (c) Set V = U + Rv then V =U 6= 0, and so V =U = R s =W with s > 0 s s and W R . The condition V =U = R =W has a regular equational presentation by (b), and the same evidently holds for W R s . (d) Again we choose an epimorphism : R q ! U and extend it to a free resolution F of R r =U . Let 'j , j = 1 . . . p be the maps in F . Then ExtiR (R r =U R ) = Im 'i = Ker 'i+1 (here denotes the R -dual). Set M = Im 'i and N = Ker 'i+1. Since the acyclicity of the resolution and the condition N = Ker 'i+1 have a regular equational presentation (for M = Im 'i this is trivial), ExtiR (R r =U R ) = N=M also has such a presentation, and an application of (b) concludes the argument. (e) One has dim R ; dim R r =U = grade R r =U since R is a regular local ring. However, grade R r =U = minfi : ExtiR (R r =U R ) 6= 0g, and the vanishing and non-vanishing of Ext can be captured by equations according to (d). m
m
.
.
339
8.4. Hochster's niteness theorem
The reduction to the ane case. The reader should note that we are free
to extend the set of indeterminates appearing in E and the system E itself. Moreover, we can also change the family X of distinguished variables that guarantees the height of the solution. We only have to make sure that the elements to which the variables X will nally specialize generate an ideal of height n. The very rst step is a routine matter: we choose a prime ideal minimal over (x) such that height = n, then we complete R with respect to the R -adic topology, and nally replace R by R^ . Because of Cohen's structure theorem A.21 we can write R as a residue class ring of a regular local ring (containing a eld), say R = S=I where I is generated by elements b = b1 . . . bs and S has a regular system of parameters a = a1 . . . ar . Extend the set of indeterminates by A = A1 . . . Ar and B = B1 . . . Bs , and modify the system E to a system Ee as follows: each equation p(X Y ) = 0 is replaced by the equation p(X Y ) = Cp1B1 + + CpsBs where the Cpj are new indeterminates. Next we enlarge Ee by further equations expressing (i) the condition that dim S=(b1 . . . bs ) = n and (ii) the fact that a power of each ai lies in the ideal generated by b and preimages of the xj . While the equations for (ii) simply exist because x is a system of parameters of R , we must invoke 8.4.4(e) for (i). Next suppose part (b) of the theorem has been proved. Then we can nd a solution ex ey to the system Ee in a regular local ring Se in which A specializes to a regular system of parameters ea. We simply set R 0 = S 0 =(b0). The original system E is solved by the residue classes x0 and y 0 in R 0 of the families ex and ye. Moreover, dim R 0 = n because of the extra equations for (i) above, and x0 is a system of parameters by the additional equations for (ii). p
p
p
p
p
p
The reduction to the ane case. For the next reduction step Artin's ap-
proximation theorem will be crucial. In order to explain it we need the theory of Henselian local rings. We must content ourselves with a very brief sketch, referring the reader to Grothendieck 142], IV, x18, Nagata 284], or Raynaud 301] for a full treatment. A local ring (R ) is Henselian if it has the following property: suppose f 2 R X ] is a monic polynomial such that its residue class f modulo R X ] has a factorization f = g0 h0 with monic polynomials g0 h0 2 (R= )X ] for which (g0 h0) = (R= )X ] then there exist monic polynomials f g 2 R X ] such that f = gh and g = g0 h = h0. A more abstract characterization is that R is Henselian if and only if every R -algebra S which is a nite R -module is a product of local rings. m
m
m
m
340
8. Big Cohen{Macaulay modules
Hensel's lemma says that a complete local ring is Henselian. Moreover, for each local ring (R ) there exists a Henselization (R h h ) which, in a sense, is the smallest Henselian local ring containing R . One has local ^ ^ ), and R h = h . The ring R h is a embeddings (R ) (R h h) (R direct limit of subrings S each of which is the localization of a module nite extension of R with respect to a maximal ideal lying over . More precisely, S has the form (R X ]=f )( X ) where f = X n + cn;1X n;1 + + c0 is a monic polynomial with c0 2 , c1 2= . (It follows that R h is a at extension of R .) We can now formulate (a special case of) Artin's approximation theorem 13]: Theorem 8.4.5. Let R be a local ring which is a localization of an ane m
m
m
m
m
m
m
m
m
m
m
E
algebra over a eld k. Let be a system of polynomial equations over R in n variables. If has a solution x R^ n, then it has a solution x0 (R h)n . Furthermore, given t, the solution x0 can be chosen such that it approximates x to order t, that is, x x0 mod t R^ n. We will not need the statement about approximation to order t it
E
2
2
m
has only been included for completeness. Proposition 8.4.6. Let E ZX Y ] be a system of equations with X = X1 . . . Xn and Y = Y1 . . . Ym . Suppose that E has a solution x y in a
regular local ring R that contains a eld K and such that x is a regular system of parameters. Then there exist an algebraically closed eld L of like characteristic, an ane domain A over L, and a maximal ideal of A with A regular such that has a solution x0 y 0 in A for which x0 is a regular system of parameters of A . Proof. We may assume R is complete. By Cohen's structure theorem A.21, R is just a formal power series ring K x]] in which the elements of x are the indeterminates. It is obviously harmless to replace K by an algebraic closure L. m
E
m
m
m
Next the indeterminates X in the system E are replaced by the elements of x so that we obtain a system of polynomial equations E0 in the unknowns Y over Lx]. By hypothesis it has a solution in the completion Lx]] of A0 = Lx](x) with respect to its maximal ideal 0 = xA0 . Thus the approximation theorem yields a solution in the Henselization of A0 , and therefore in an extension A00 = (A0X ]=(f ))( 0 X ) where f = X n + cn;1X n;1 + + c0 is a monic polynomial with c0 2 0 , c1 2= 0 . It is easily veri ed that dim A00 = n and that the image of x generates the maximal ideal of A00. It follows that A00 is a regular local ring for which x is a regular system of parameters. In order to arrive at an integral domain A we replace A0X ]=(f ) by the residue class ring with respect to its unique minimal prime ideal contained in ( 0 X )A0X ]=(f ). m
m
m
m
m
341
8.4. Hochster's niteness theorem
The nal step. Let L, A, , x0 , and y 0 be as in 8.4.6. We rename x0 and y 0 by setting x = x0 , y = y 0. Since L is algebraically closed, the injection L A induces an isomorphism L = A= by Hilbert's Nullstellensatz in its algebraic version see A.15. In other words, A = L as an L-vector space. This implies that A = Lz1 . . . zr ] where z1 . . . zr generates the ideal . We write A = LZ ]=I , Z = Z1 . . . Zr being indeterminates over L. The ideal I is nitely generated by polynomials f1 . . . fs . Let C L m
!
m
m
m
be the nite set of coecients appearing in (1) the polynomials f1 . . . fs , (2) n + m polynomials expressing x1 . . . xn and y1 . . . ym in terms of z1 . . . zr , P (3) a polynomial g 2= and polynomials hij such that gzi = hij xj for all i. (The polynomials hij can be found since x generates A .) Let L0 be the sub eld generated by C over the prime eld of L and set R 0 = L0 z1 . . . zr ]. The rst point to be observed is that I \ L0Z ] is generated by f1 . . . fs , since the extension L0 Z ] ! LZ ] is faithfully at. This implies R 0 = L0Z ]=(f1 . . . fs )L0Z ] therefore A = L L R 0. Obviously z1 . . . zr generate a maximal ideal 0 of R 0, and R 0 = 0 = L0 hence R 0 = L0 (z1 . . . zr )R 0. The extension R 0 0 ! A is at, and so dim R 0 0 = dim A = n by A.11. The solution x y is contained in R 0 , and 0 R 0 0 is generated by x because of (3) above. Let us rst treat the case of characteristic 0 after all, the descent from characteristic 0 to positive characteristic is the main point of 8.4.1. (In positive characteristic it only remains to replace L0 by a nite eld of the same characteristic.) With the notation just introduced, L0 is the eld of fractions of the nitely generated Z-algebra B = ZC ], and apart from the fact that the coecients no longer form a eld, almost nothing is lost if we replace L0 by B . (i) R 00 = B z1 . . . zr ] is a domain containing x y . This is obvious. (ii) = (z1 . . . zr )R 0 \ R 00 is a prime ideal of height n: evidently is a prime ideal, and it has the same height as its extension in R 0, a ring of fractions. (iii) = (z1 . . . zr )R 00 and R 00 = B . This is an immediate consequence of R 0 = L0 (z1 . . . zr )R 0. (iv) R 00 is generated by x because C contains all the coecients appearing in (3) above and g 2= . For the very last step we choose a maximal ideal of B not containing the constant coecient of the polynomial g appearing in (3) above. This is possible since Z and, hence, B are Hilbert rings, and for the same reason B= is a nite eld (see A.18). We claim that replacing all the data by their residue classes mod gives us the desired solution of E in m
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a local ring of characteristic p. First note that x generates since is generated by the residue classes zi of the zi , and these in turn can be written as linear combinations of the x j because the polynomial g of (3) above is non-zero modulo . Second, height = n as will now be shown. Let = . Then height height +dim(R 00 = R 00 ) by A.5, applied to the homomorphism B ! R 00 , and therefore height = dim(R 00 = R 00 ) height ; height : The ring R 00, a residue class ring of a polynomial ring over Z, and thus of a Cohen{Macaulay ring, is catenary see 2.1.12. Since R 00 is also a domain, height = height + height = = height + height or height height = n, as desired. In positive characteristic the argument is essentially the same: one only has to replace Z by the prime eld of L0. p
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Exercise 8.4.7. Let R be a regular local ring and U , M , N nite R -modules, given as quotients of nite free R -modules by submodules. Show that both the acyclicity and the non-acyclicity of a complex U ! M ! N have regular equational presentations. (Describe the maps by matrices.)
8.5 Balanced big Cohen{Macaulay modules Big Cohen{Macaulay modules M lack many of the properties of nite Cohen{Macaulay modules. For example, let R = K X Y ]] and M = R Q, where Q is the eld of fractions of R=(Y ). Then X is obviously regular on M , and M=XM = R=(X ). Thus M is (X Y )-regular, but not (Y X )-regular. However, it is important for the applications in Chapter 9 and an interesting fact in itself that every local ring R possessing a big Cohen{Macaulay module even has a balanced big Cohen{Macaulay module, i.e. a module M such that every system of parameters is an M sequence. More precisely, we want to prove that the -adic completion of any big Cohen{Macaulay module is balanced. Our main argument will be that 1.1.8 has a converse for complete modules: Theorem 8.5.1. Let R be a ring, x = x1 . . . xn a sequence of elements of R, and M an R-module. Let I = xR, and denote the I-adic completion of ^ Then the following are equivalent: M by M. (a) x is M-quasi-regular ^ (b) x is M-quasi-regular ^ (c) x is M-regular. m
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By de nition quasi-regularity includes the requirement IM 6= M . ^ j +1M^ is naturally isomorphic with I j M=I j +1M , one has a Since I j M=I commutative diagram M (R=I )X1 . . . Xn] ;;;;! grI M
Proof.
M^
(R=I )X1 . . . Xn] ;;;;! grI M^
Together with the description of quasi-regularity by the conclusion of 1.1.8 this diagram immediately yields the equivalence of (a) and (b). Theorem 1.1.8 says that (c) implies (b). We want to prove the crucial implication (b) ) (c) by induction on n, and recall the results of Exercise 1.1.15, namely (i) if x1 z 2 I i M for z 2 M , then z 2 I i;1 M , (ii) the sequence x2 . . . xn is (M=x1 M )-quasi-regular. T Let z 2 M such that x1z = 0. Then, by (i), z 2 I j M^ = 0, and hence x1 is ^ -regular. Because of (ii) it remains to prove that M=x ^ 1 M^ ^ 1M^ )b. M = (M=x There is a natural exact sequence ^ 1M^ )b ;! 0 0 ! (x1M^ )0 ;! M^ ;! (M=x in which (x1M^ )0 is the completion of x1M^ with respect to its subspace topology (see 270], Theorem 8.1 note that the quotient topology on ^ 1M^ is just the I -adic topology). The subspace topology is given M=x by the ltration (x1 M^ \ I j M^ ). Of course x1 M^ = M^ is complete in its own I -adic topology, and we are left to verify the following claim of Artin{Rees type: if x is M -quasi-regular for some R -module M , then x1M \ I j M I j ;1x1M . But this follows immediately from (i). Since quasi-regularity of a sequence is invariant under permutations of its elements, one can permute M^ -regular sequences: Corollary 8.5.2. With the notation of 8.5.1 assume that x = x1 . . . xn is ^ M-regular. Then for every permutation of f1 . . . ng the sequence x = ^ x(1) . . . x(n) is M-regular. Another consequence is the existence of balanced big Cohen{Macaulay modules: Corollary 8.5.3. Let (R ) be a Noetherian local ring, and M a big ^ is a balanced Cohen{Macaulay R-module. Then the -adic completion M m
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big Cohen{Macaulay module. In particular, if R contains a eld, it has a balanced big Cohen{Macaulay module.
Note that for any system of parameters x the -adic and (x)-adic topologies on M coincide. Therefore we can apply 8.5.1 to the -adic completion. Proof.
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Suppose that the system of parameters x = x1 . . . xn is an M sequence, and let y = y1 . . . yn be an arbitrary system of parameters. By the standard prime avoidance argument there exists an element w 2 not contained in any minimal prime ideal of (x1 . . . xn;1) or (y1 . . . yn;1). Hence x1 . . . xn;1 w and y1 . . . yn;1 w are systems of parameters. Note that x1 . . . xn;1 w is an M -sequence: a power of xn being a multiple of w modulo (x1 . . . xn;1), the element w must be regular on M=(x1 . . . xn;1)M . Then w x1 . . . xn;1 is M -quasi-regular and therefore ^ -regular. Furthermore M=w ^ M^ is an (x1 . . . xn;1)-regular module for M the local ring R = R=Rw. By induction on n one may assume that ^ M^ )-regular, too. Then w y1 . . . yn;1 is M^ -regular, ( y1 . . . y n;1) is (M=w and applying the preceding arguments in reverse order we get that y is an M^ -sequence. Now the second part of the corollary follows immediately from the existence of big Cohen{Macaulay modules for local rings containing a eld see 8.4.2. Remark 8.5.4. A di erent construction of balanced big Cohen{Macaulay modules was given by Grith in 139] and 140]. Let R be a Noetherian complete local domain containing a eld K . As in the proof of 8.4.6, R is a module- nite extension of a formal power series ring K x]] where x is an arbitrary system of parameters. By 139], Theorem 3.1, there exists an R -module which is a free A-module (with countable basis). Such a module is a balanced big Cohen{Macaulay module (140], Proposition 1.4). Balanced big Cohen{Macaulay modules are much closer to nite modules than is apparent from their de nition. Sharp 342] developed the theory of grade for balanced big Cohen{Macaulay modules, similar to that for nite modules, using Theorem 8.5.6 below as a prime avoidance argument. Unfortunately, however, the property of being a balanced big Cohen{Macaulay module is not stable under localization. In Chapter 9 we shall introduce a general notion of grade that overcomes this obstacle. Proposition 8.5.5. Let R be a Noetherian local ring, and M a balanced big m
Cohen{Macaulay module. (a) One has dim R= = dim R for all Ass M (b) in particular Ass M is nite (c) Ass M consists of the minimal prime ideals of Ann M, and so Supp M = V (Ann M ). Proof. (a) Let Spec R with dim R= < dim R . Then is not contained in a prime ideal such that dim R= = dim R , hence not contained in the union of these prime ideals. So there exists x with dim R=(x) < dim R . The element x can be extended to a system of parameters. Thus it is regular on M , and = Ass M . p
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Part (b) follows immediately from (a). (c) Let Ass M = f 1 . . . r g. Since Ann M i for all i, it remains T to be shown for the rst assertion in (c) that f j 2 Ann M for all f 2 ri=1 i and j 0. Given such an element f , there exists j with f j =1 = 0 in R i for all i. It follows that f j M i = 0, and so i 2= Ass fM . On the other hand, Ass f j M Ass M . Therefore Ass f j M = , which is only possible if f j M = 0. The second assertion in (c) follows from the rst since, over a Noetherian ring, every prime ideal 2 Supp M contains a 2 Ass M . Theorem 8.5.6. Let R be a Noetherian local ring, and M a balanced big Cohen{Macaulay module. Suppose that x = x1 . . . xr is an M-sequence. Then Ass M=xM is nite, and dim R= = dim R ; r for all 2 Ass M=xM. Proof. Let be prime ideal such that dim R= = dim R . Then Rx1 + Ann M 6 : if 2 Ass M , then x1 2= if 2= Ass M , then Ann M 2= by 8.5.5. As the number of such prime ideals is nite, Rx1 + Ann M is not contained in their union. By 1.2.2 one therefore nds an element y 2 Ann M with x1 + y 2= for all such that dim R= = dim R . The following facts are now obvious: (i) dim(R=(x1 + y)) = dim R ; 1, and M=(x1 + y)M is a balanced big Cohen{Macaulay module over R=(x1 + y) (ii) x1 + y x2 . . . xr is an M -sequence (iii) M=xM = M=(x1 + y x2 . . . xr )M . = M=(x1 + y)M . Because of (i) and (ii) we Set R = R=(x1 + y) and M -sequence x 2 . . . x r . By (iii) the can apply an inductive argument to the M associated primes of M=xM are exactly the preimages of the associated ( x2 . . . x r )M over R . primes of M= p
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Exercises Prove that each of the conditions (a), (b), and (c) of 8.3.5 is equivalent to
8.5.7.
^ being a (balanced) big Cohen{Macaulay module. M 8.5.8. Let R be a Noetherian local ring, and M a balanced big Cohen{Macaulay module over R . One sets supp M = Spec R : M = M = 0 . Show (a) M is a big Cohen{Macaulay module for R if and only if supp M , (b) one has height + dim R = dim R for every supp M . (For general M one uses 8.5.9(c) to de ne supp M see Foxby 116].) 8.5.9. Let R and M be as in 8.5.8. Then verify that the following are equivalent for Spec R : (a) supp M (b) there exists an M -sequence a1 . . . ar with Ass(M=(a1 . . . ar )M ) (c) there exists i with i( M ) = 0 (d) there exists i with H i R (M ) = 0 (e) H i R (M ) = 0 for i = 0 . . . height 1 and H hR (M ) = 0 for h = height .
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Hint: (b) ) (c): localize and consider i = r (c) ) (d): use 3.5.12 (d) ) (e): by hypothesis on M there exists an M -sequence a1 . . . ah in (e) ) (a): this holds for arbitrary M by 3.5.6. p
Notes The results in this chapter on the existence of big Cohen{Macaulay modules are entirely due to Hochster, as well as the method of their construction. With one exception we have followed closely Hochster's original treatment in 178] and his inuential lecture notes 181]. The exception is the existence of `amiable' systems of parameters, for which Hochster avoids local cohomology. The results 8.1.3, 8.1.4, and 8.1.5 are due to Schenzel 326], 329]. The proof of Roberts' theorem 8.1.2 is a slight variation of his original argument 309] introduced by Schenzel 328] in order to obtain a somewhat more general result. There have been suggestions for modifying Hochster's methods. Bartijn and Strooker 364] constructed a `pre{Cohen{Macaulay' module by `monomial modi cations', and showed that the -adic completion of such a module is a balanced big Cohen{Macaulay module. Our proof of the existence of such modules is a variant of their arguments, whereas the rst construction was given by Hochster in 180] (based on an extension of the `modi cation method'). Grith's work was mentioned in 8.5.4. There are several articles which deal with the properties of balanced big Cohen{Macaulay modules: see Duncan 78], Sharp 342], 343], Zarzuela 398] and the literature quoted in these papers. Theorem 8.5.6 and Exercises 8.5.8 and 8.5.9 have been taken from Sharp 342], who coined the notion `balanced' in that article. A very interesting revision of Hochster's arguments is due to van den Dries who introduced methods of model theory to our subject see Chapter 12 of Strooker 364]. A completely di erent construction in characteristic zero was given by Roberts 310] who derived the existence of a `Cohen{Macaulay complex' from resolution of singularities and the Grauert{Riemenschneider vanishing theorem. We refer the reader to Hochster and Huneke 197] for a more extensive list of properties that have a regular equational presentation. It is still open whether there exist big Cohen{Macaulay modules for local rings of mixed characteristic. The most intensive attempts towards their construction can be found in Hochster's article 180]. We saw in 2.1.14 that one cannot expect a nite maximal Cohen{ Macaulay module for every local ring. Of course a local ring of dimension 1 has such a module, and also a complete local ring of dimension 2 (2.1.20 and 2.2.31). It is an open question whether there exist nite maximal Cohen{Macaulay modules for all complete local rings. A very special positive result in dimensions > 2 is due to Hartshorne, Peskine, m
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and Szpiro see 180], (5.12). Hochster's article 180] also contains a discussion of the question whether there exist big Cohen{Macaulay algebras. For positive characteristics Hochster and Huneke 193] have answered this question: if R is excellent, then the integral closure of R= where is a prime ideal with dim R= = dim R in an algebraic closure of its eld of fractions is a big Cohen{Macaulay module for R (see also Remark 10.1.6). The material on the Frobenius functor has been taken from Peskine and Szpiro's ingenious thesis 297]. Theorem 8.2.7 is just the rst in their long series of surprising results, many of which will be dealt with in Chapter 9. Kunz's characterization 245] of regular local rings of characteristic p and Herzog's converse 157] of 8.2.7 were mentioned in 8.2.11. More recently, the Frobenius functor was investigated by Dutta 79], 83] and Seibert 331]. p
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This chapter is devoted to the consequences of the existence of big Cohen{Macaulay modules for local rings containing a eld. Among the theorems covered, the reader will nd Hochster's direct summand theorem for regular local rings, his canonical element theorem, the Peskine{Szpiro intersection theorem and its extensions, the theorem of Evans and Grith on ranks of syzygy modules, and, nally, bounds for the Bass numbers of modules. These bounds entail surprising characterizations of Cohen{ Macaulay and Gorenstein local rings. There exist derivations of all the theorems in this chapter avoiding big Cohen{Macaulay modules most of them will only be outlined briey. They were found in attempts to prove the theorems in mixed characteristic. With the main exception of Roberts' new intersection theorem (whose proof in mixed characteristic requires methods beyond the scope of this book) these e orts have not yet succeeded. 9.1 Grade and acyclicity The fundamental argument in Sections 9.4{9.6 is that certain nite free complexes become exact when tensored with a balanced big Cohen{ Macaulay module. This section contains the acyclicity criterion on which our treatment is based. Let R be a Noetherian ring, F. : 0
' ' ;! Fs ;! Fs;1 ;! ;! F1 ;! F0 ;! 0 s
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a complex of nite free R -modules, and M an R -module. We saw in 1.4.13 thatPF is acyclic if and only if grade Iri ('i ) i for i = 1 . . . s. Here ri = sj =i(;1)j ;i rank Fj is the expected rank of 'i . Now we want to develop a more general criterion by which one can decide whether F M is acyclic for a given R -module M . As we shall see, the condition grade Iri ('i) i is just to be replaced by grade(Iri ('i ) M ) i. It will be crucial that we can use the general criterion for a balanced big Cohen{ Macaulay module M . Therefore we must rst introduce a concept of grade which does not exclude non- nite modules. De nition 9.1.1. Let R be a ring, I an ideal generated by x = x1 . . . xn , and M an R -module. If all the Koszul homology modules Hi (x M ) .
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vanish, then we set grade(I M ) = 1 otherwise grade(I M ) = n ; h where h = supfi : Hi (x M ) 6= 0g. Note that by 1.6.22 grade(I M ) is well de ned: it does not depend on the choice of x. Furthermore, 1.6.17 shows that for a nite module M over a Noetherian ring R the de nition of grade is consistent with that in Chapter 1. There is not much point in considering non- nite ideals I for completeness let us de ne grade(I M ) to be the supremum of grade(I 0 M ) where I 0 ranges over the nitely generated subideals of I . (This makes sense because grade is monotone with respect to inclusion of ( nite) ideals see 9.1.2.) On the other hand, there is no reason to restrict ourselves to Noetherian rings, as we shall see below. Proposition 9.1.2. Let R be a ring, I a nite ideal, and M an R-module. Then
(a) grade(I M ) = 0 () HomR (R=I M ) 6= 0 () fz 2 M : Iz = 0g 6= 0 (b) if y = y1 . . . ym is a weak M-sequence in I, then grade(I M ) m, and grade(I M=y M ) = grade(I=(y ) M=y M ) = grade(I M ) ; m (c) if R ! S is a at ring homomorphism, then grade(IS M S ) grade(I M ) in particular grade(I M ) grade(I M ) for 2 Spec R (d) if R ! S is faithfully at, then grade(IS M S ) = grade(I M ) (e) if 0 ! U ! M ! N ! 0 is an exact sequence of R-modules, then grade(I M ) minfgrade(I U ) grade(I N )g grade(I U ) minfgrade(I M ) grade(I N ) + 1g grade(I N ) minfgrade(I U ) ; 1 grade(I M )g (f) if J I is nite, then grade(J M ) grade(I M ) (g) if S is a subring of R containing a system of generators x of I, then grade(xS M ) = grade(I M ). Proof. (a) By virtue of 1.6.16 one has Hn (x M ) = HomR (R=I M ) for every system of generators x = x1 . . . xn of I . (b) The inequality grade(I M ) m follows immediately from 1.6.16. Let denote residue classes modulo (y ). We have an isomorphism see 1.6.7. This shows K (x) R M = K (x) R R R M = K ( x) R M grade(I M ) = grade(I M ). Now we extend y by a sequence z to a system ) = grade(I M ) ; m by 1.6.13, and of generators of I . Then grade((z ) M ) follows as above. grade((z ) M ) = grade(( z ) M ) = grade( I M (c) and (d) are immediate consequences of 1.6.7. (e) One argues as in the proof of 1.2.9, but uses the exact sequence 1.6.11 of Koszul homology rather than that of Ext. (To carry the analogy one step further, one could work with Koszul cohomology see 1.6.10.) (f) It is enough to consider the case in which J = I + (y). Let I = (x) and compare H (x M ) and H ((x y) M ) via 1.6.13. p
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(g) Let xS denote x as a sequence in S . By 1.6.7 one has K (xS ) S M = K (xS ) S R R M = K (x) R M . Part (g) of 9.1.2 explains why the computation of grade can always be reduced to a situation in which R is Noetherian: one simply replaces R by the Z-subalgebra generated by a system of generators of I . For inductive proofs one must be able to decrease grade by passing to residue classes modulo a regular element. In general, one cannot nd such an element in an ideal of positive grade, but one need not go very far. For simplicity we write I X ] for IR X ] and M X ] for M R X ]. Proposition 9.1.3. Let R be a ring, I and J nite ideals, and M an R-module. (a) Suppose grade(I M ) 1. Then I X ] contains an M X ]-regular element. (b) One has grade(IJ M ) = min(grade(I M ) grade(J M )). Proof. (a) We may replace R by a Noetherian subring. Let x1 . . . xn generate I , and set y = x1 + x2X + + xn X n;1. If y is a zero-divisor, then it is contained in an associated prime of M X ], whether M is nite or not. But M X ] is a graded module over the graded ring R X ], so by 1.5.6 y annihilates a non-zero homogeneous element of M X ] which necessarily has the form X pz , z 2 M , z 6= 0. It follows that Iz = 0 which contradicts our hypothesis. (b) We go by induction on grade(IJ M ). If grade(IJ M ) = 1, then the assertion follows from 9.1.2(f). If grade(IJ M ) = 0, then grade(I M ) = 0 or grade(J M ) = 0 by 9.1.2(a). In the other case we may rst adjoin an indeterminate because of 9.1.2(d). Then IJ contains an M -regular element y by (a) now one replaces all data by their residue classes modulo y, and applies the induction hypothesis in conjunction with 9.1.2(b). It remains to add a proposition which describes the special properties of grade over Noetherian rings R for these it makes sense to introduce the notation depth M = grade( R M ): Proposition 9.1.4. Let R be a Noetherian ring, I an ideal in R, and M an .
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R-module. Then (a) grade(I M ) = 0 if and only if there exists Ass M with I , (b) Ass M depth M = 0, (c) grade(I M ) = inf depth R : V (I ) . Proof. (a) One has grade(I M ) = 0 exactly when there exists a non-zero x M such that Ix = 0. Over a Noetherian ring, I must be contained in an associated prime ideal of M . (b) Because of (a), depth R = 0 is equivalent with R Ass M , and this holds if and only if Ass M . (One only needs that is nite.) (c) One has grade(I M ) depth M for all V (I ). So (c) is trivial when grade(I M ) = . Suppose that in the case of nite grade we have p
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found 2 V (I X ]) such that depth M X ] = grade(I X ] M X ]), and set = R \ . Then depth M = grade( R M ) = grade( R X ] M X ] ) depth M X ] : Together with grade(I X ] M X ]) = grade(I M ) this yields grade(I M ) = depth R , and thus the assertion. In order to nd one proceeds by induction, using the fact that I X ] contains an M -regular element if grade(I M ) > 0 the case of grade zero is covered by (a) and (b). Let ' : F ! G be a homomorphism of nite free R -modules, and M 6= 0 an R -module. We say the ' has rank r with respect to M , if grade(Ir (') M ) 1, whereas Ir+1(')M = 0. We write rank(' M ) = r. Note that rank(' M ) may not be de ned. Furthermore, rank(' R ) = rank ' by 1.4.11. For systematic reasons one sets rank(' M ) = 0 when M is the zero module. Proposition 9.1.5. Let R be a ring, M 6= 0 an R-module, and F : 0 ! Fs ! Fs;1 ! ! F1 ! F0 ! 0 a complex of nite free R-modules such that F M is acyclic. Let 'i denote the map Fi ! Fi;1. Then P rank('i M) is the expected rank ri of 'i for i = 1 . . . s: rank('i M ) = sj =i(;1)j ;i rank Fj . Proof. We choose bases of the free modules, and matrices Ai representing the homomorphisms 'i. Let S be the Z-subalgebra generated by the entries of all these matrices. They de ne a complex F 0 of nite free S -modules such that F 0 S R = F . Therefore F 0 S M = F R M is acyclic. The ring S is Noetherian. For 2 AssS M the complex F 0 S is split exact by 1.4.12 which furthermore implies that Iri (Ai) = S and Iri+1(Ai) = 0. Since S is Noetherian, one has grade(Iri (Ai) M ) 1 by 9.1.4. Let I = Iri +1(Ai). Assume IM 6= 0, and choose z 2 M such that Iz 6= 0. Then Ass Iz 6= . For 2 Ass Iz one has (Iz ) 6= 0, and hence IM 6= 0, which is a contradiction: since Ass Iz Ass M , one even has I = 0 as seen above. It follows that rank(Ai M ) = ri . By the de nition of rank and 9.1.2 it is irrelevant for rank(Ai M ) whether one considers Ai as a matrix over S or R . Theorem 9.1.6 (Buchsbaum{Eisenbud Northcott). Let R be a ring, M P
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The remark about ri that follows 1.4.13 applies here, too: each of (a) and (b) implies that ri 0 for i = 1 . . . s. Proof. As in the proof of 9.1.5 one reduces the theorem to the case in which R is Noetherian. Then the proof is mutatis mutandis the same as that of 1.4.13. We indicate some of the modi cations. There is nothing to prove if M = 0, so assume that M is non-zero. For (a) ) (b) one uses 9.1.5 to get grade(Iri ('i) M ) 1. Then one adjoins an indeterminate, which a ects neither the acyclicity of the complex nor the grades under consideration. By virtue of 9.1.3 one nds an M -regular element in the intersection of the ideals Iri ('i ), and completes the proof of (a) ) (b) as in the case of 1.4.13. It is not necessary to pass from R to R=(x) instead one substitutes M=xM for M in order to apply the induction hypothesis. If xM = M , then grade((x) M ) = grade(Iri ('i ) M ) = 1 for all i. For (b) ) (a) one sets Mi = Coker 'i+1 M , and replaces depth R by depth M , and Fi by Fi M . That (Mi ) is free for depth R i, must be modi ed to `(Mi ) is a direct sum of nitely many copies of M if depth M i'. We introduce a new invariant of a complex and provide a lemma which is fundamental for the results in Sections 9.4{9.6. (Recall that codim I = dim R ; dim R=I for an ideal I in a local ring R .) De nition 9.1.7. Let (R ) be a Noetherian local ring, p
p
p
p
p
p
p
m
F. : 0
' ' ;! Fs ;! Fs;1 ;! ;! F1 ;! F0 ;! 0 s
1
a complex of nite free R -modules, and ri the expected rank of 'i. We de ne the codimension of F by .
codim F = inf fcodim Iri ('i) ; i : i = 1 . . . sg: .
If F is acyclic, then codim F 0 by the Buchsbaum{Eisenbud acyclicity criterion 1.4.13 (or 9.1.6) since grade I codim I for all ideals. Conversely, if codim F 0, then F need not be acyclic, but F M is acyclic for a balanced big Cohen{Macaulay module M : Lemma 9.1.8. Let (R ) be a Noetherian local ring, and F a complex of nite free R-modules as above. Suppose that codim F 0. Then F M .
.
.
.
.
.
m
.
is acyclic for every balanced big Cohen{Macaulay module M.
.
In view of 9.1.6 it is enough that grade(Iri ('i) M ) i for i = 1 . . . s. In fact, if I is an ideal with codim I i, then it contains a sequence x1 . . . xi which is part of a system of parameters, as is easily shown by induction on i. Such a sequence is M -regular.
Proof.
353
9.1. Grade and acyclicity
In Section 9.5 we shall investigate lower bounds for the numbers ri . A rst result in this direction can be recorded already. Let (R ) be a local ring. We say that a complex of nite free R -modules F as in 9.1.7 is minimal of length s if Fs 6= 0 and 'i (Fi ) Fi;1 for all i. Considering minimal complexes only is not a severe restriction since every complex of nite free modules over a local ring decomposes into a direct sum of a minimal such complex and a split exact one. Proposition 9.1.9. Let (R k) be a local ring, and F a length s minimal m
.
m
.
m
complex of nite free R-modules as above. Suppose there exists an Rmodule M such that M = M and F. M is acyclic. Let ri denote the expected rank of 'i . Then ri 1 for i = 1 . . . s.
6
m
One has rs = rank Fs 1 by hypothesis, and it follows from Proposition 9.1.5 that ri = rank('i M ) 0 for all i. Arguing inductively, we must only show r1 = 0 implies r2 = 0. If r1 = rank('1 M ) = 0, then I1('1 )M = 0, and so '1 M = 0. Therefore we have an exact sequence Proof.
' M M ;;;! F1 M ;! 0: Consequently F2 M k ! F1 M k ! 0 is also exact. By hypothesis M 6= M , equivalently, M k is a non-zero k-vector space. Thus the sequence F2 k ! F1 k ! 0 must be exact. On the other hand, '2 k = 0 since '2(F2 ) F1 . Hence we get F1 = 0, and r2 = rank F1 ; r1 = 0. 2
F2
m
m
Exercises Let R be a ring, I a nitely generated ideal, and M an R -module. Furthermore let R be a polynomial ring over R in an in nite number of indeterminates, I = IR , and M = M R . Prove the following: (a) If grade(I M ) < 1, then every maximal weak M -sequence in I has length equal to grade(I M ). (b) One has grade(I M ) = 1 if and only if I contains an in nite weak M sequence. (c) One has grade(I M ) = inf fi : ExtiR1 (R =I M ) 6= 0g. (d) For Noetherian R one has grade(I M ) = inf fi : ExtiR (R=I M ) 6= 0g. (e) Suppose that the number of associated prime ideals of M=(x)M is nite for every weak M -sequence x. (For example, this holds when M is a balanced big Cohen{Macaulay module over a Noetherian local ring see 8.5.6). Then one can drop the subscript 1 in (a), (b), and (c). 9.1.11. For a nite module M over a Noetherian ring R we have grade(I M ) = 1 if IM = M . For non- nite M this may be false. Find an example. 9.1.12. Let (R ) be a Noetherian local ring, and M an R -module. Prove (a) if M 6= M , then depth M is nite, 9.1.10.
1
1
1
1
1
1
1
1
m
m
1
1
1
1
354
9. Homological theorems
(b) if depth M is nite, then depth M dim R , (c) depth M = inf fi : H i (M ) 6= 0g, (d) depth M = dim R () M^ is a (balanced) big Cohen{Macaulay module. 9.1.13. Sometimes it may be more natural to work with homology modules rather than the ideals Iri ('i ). Therefore it is worth while reformulating the crucial condition for acyclicity. One must however use the homology of F. = HomR (F. R ). With the notation of 9.1.6, show the following are equivalent: (a) grade(Iri ('i ) M ) i for i = 1 . . . s (b) grade(Ann H i (F. ) M ) i for i = 1 . . . s. 9.1.14. Generalize the `lemme d'acyclicite' 1.4.24 to the case of arbitrary R -modules Li. 9.1.15. Let R be a Noetherian local ring and M a balanced big Cohen{Macaulay module. Prove TorRi (N M ) = 0 for all nite R -modules N and i > 0. In particular, M is faithfully at if R is regular. m
9.2 Regular rings as direct summands Let R and S be Noetherian local rings such that R S and S is a nite R -module. Suppose that R is regular and, for the moment, S is a Cohen{Macaulay ring. Since every system of parameters of R is a system of parameters of S , the R -module S , having nite projective dimension, must be free. Furthermore the element 1 2 R is part of an R -basis of S , and it follows that R is a direct summand of S as an R -module. Quite surprisingly this holds true regardless of the Cohen{Macaulay property of S , at least when S contains a eld. The argument above uses the fact that a system of parameters of R is an S -sequence. As we shall see, a much weaker property suces. It is given by the following `monomial theorem': Theorem 9.2.1. Let S be a Noetherian local ring containing a eld. Then for every system x = x1 . . . xn of parameters and all t 0 one has xt1 xtn 2= (xt+1). Proof. By 8.4.2 there exists an x-regular module M . Suppose that xt1 xtn 2 (xt+1). Then xt1 xtnM xt+1M . The associated graded module gr(x) M is an (R=(x)X1 . . . Xn])-module in a natural way, and, a fortiori, X1t Xnt gr(x) M (X1t+1 . . . Xnt+1) gr(x) M . On the other hand, since x is an M -sequence, the associated graded module gr(x) M is isomorphic to M R=(x)X1 . . . Xn] (see 1.1.8). Therefore M (gr(x) M ) (X1t+1 . . . Xnt+1) gr(x) M = X1e Xnen (M=xM ) as an R -module where the direct sum is taken over all monomials X1e Xnen 2= (X1t+1 . . . Xnt+1): This is a contradiction since X1t Xnt 2= (X1t+1 . . . Xnt+1). 1
1
355
9.2. Regular rings as direct summands
The proof of 9.2.1 shows much more than stated in the theorem: let I0 J0 be ideals in ZX1 . . . Xn] generated by monomials, and I1 J1 the ideals generated by the corresponding monomials in x1 . . . xn then I1 J1 () I0 J0. Suppose that S = R C as an R -module. Then for every ideal I R one has IS = I IC , hence IS \ R = I . Let x = x1 . . . xn be a system of parameters of R . If R is regular, then, as the proof of 9.2.1 shows, xt1 xtn 2= (xt+1) so xt1 xtn 2= xt+1S , for otherwise xt1 xtn 2 (xt+1)S \ R = (xt+1). This simple observation proves the easy part of the following lemma. Lemma 9.2.2. Let (R ) be a regular local ring and x = x1 . . . xn a m
regular system of parameters. Suppose that S R is an R-algebra which is nite as an R-module. Then R is a direct R-summand of S if and only if xt1 xtn = xt+1S for every t 0.
2
Since the -adic completion R^ is a faithfully at extension of R , the same holds true for the extension S R^ of S . Thus xt1 xtn 2= xt+1S implies that xt1 xtn 2= xt+1S R^ . Suppose that the implication still open holds under the additional assumption that the regular local ring is complete. Then R^ is a direct R^ summand of S R^ and the natural homomorphism (given by restriction of maps) ^ R^ ) ;! HomR^ (R ^ R^ ) ^ : HomR^ (S R is surjective. Since S is a nitely presented R -module, one has a natural commutative diagram Proof.
m
HomR (S R ) R^
R^ ;;;;! HomR (R R ) R^
^ ^ R^ ) ;;;;! ^ R^ ) HomR^ (S R HomR^ (R where : HomR (S R ) ! HomR (R R ) is again given by restriction. Since R^ is surjective, itself must be surjective: the identity map on R can be extended to an R -homomorphism S ! R , so R is a direct R -summand of S . After these preparations we may assume that R is complete. Let Rt = R=(xt), and St = S=xt S Rt is a Gorenstein ring of dimension zero. Since xtn;1 xtn;1 2= xt, but xt1;1 xtn;1 xt , the residue class of xt1;1 xtn;1 generates Soc Rt . Therefore each of the induced maps 't : Rt ! St is injective otherwise its kernel would contain Soc Rt, whence xt1;1 xtn;1 2 xtS , contradicting the hypothesis of the lemma. Furthermore Rt is an injective Rt -module. Thus each of the maps 't splits: there is an Rt homomorphism t : St ! Rt such that t 't = idRt . m
356
9. Homological theorems
The ideals (xt ) form a system co nal with that of the powers of . Since R is -adically complete, one has m
m
HomR (S R ) = HomR (S lim ; Rt) = lim ; HomR (S Rt) = lim ; HomRt (St Rt):
In the latter inverse system the map ij : HomRi (Si Ri) ! HomRj (Sj Rj ) associates to each homomorphism Si ! Ri the induced map Sj ! Rj . We have to nd homomorphisms t : St ! Rt such that (i) ij (i ) = j , and (ii) t 't = idRt . Because of (i) we then obtain a homomorphism lim t = : S ! R , which by (ii) satis es jR = idR : if (y) 6= y for ; some y 2 R , then t 't 6= idRt for every t such that (y) ; y 2= (xt ). Let Dt be the set of homomorphisms : St ! Rt for which 't = idRt . Obviously the sets Dt HomRt (St Rt) are non-empty and form an inverse system. However, since the maps ij jDi : Di ! Dj may not be surjective, we cannot immediately conclude that lim Dt 6= , as desired. Instead we ; de ne subsets \ Et = it(Di ): i t
Then Et = it(Ei ) for all i with i t, and it is enough to show that Et 6= for some, equivalently all, t. Every Di is an ane subspace of HomRi (Si Ri) that is, it is of the form i + Ui with a submodule Ui . Therefore tt(Dt )
t+1t(Dt+1) it(Di)
is a decreasing chain of non-empty ane subspaces Aj of HomRt (St Rt ). Consequently the submodules Mj = f ; : 2 Aj g are non-zero and form a decreasing chain, too. This chain stabilizes in the Artinian module HomRt (St Rt), and so does the chain of ane subspaces Aj . A consequence of 9.2.1 and 9.2.2 is the `direct summand theorem' for regular local rings: Theorem 9.2.3 (Hochster). Let R be a regular local ring containing a eld,
and S R an R-algebra which is a nite R-module. Then R is a direct summand of the R-module S.
As in the proof of the lemma, we may assume that R is complete. Let be a prime ideal of S lying over the zero-ideal of R . If R is a direct R -summand of S= , then it is a direct R -summand of S : compose a section of the natural embedding R ! S= with the natural epimorphism S ! S= . Being an integral domain which is module- nite over a complete local ring, S= is local itself (284], (30.5)), and we can invoke 9.2.1 and 9.2.2. Proof. p
p
p
p
p
357
9.3. Canonical elements in local cohomology modules
Remarks 9.2.4. (a) In characteristic zero a much weaker property than regularity is sucient for the direct summand property of R as described by 9.2.3. Let R be a Noetherian normal domain containing a eld of characteristic zero, and S a module- nite extension ring. In showing that R is a direct R -summand of S , it is harmless to replace S by any S -algebra T (see the proof of 9.2.3). So we rst factor out a prime ideal of S lying over the zero-ideal of R , and may assume that S is a domain. Then we extend the eld of fractions of S to a nite normal extension L of the eld K of fractions of R , and replace S by the integral closure T of R in L. Let d = dimK L and Tr: L ! K denote the trace map. Then for every x 2 K one has (1=d ) Tr x = x, and Tr y 2 R for every y 2 T , since the trace of an integral element is integral and R is integrally closed in K . (We refer the reader to 397], Chapter II for the eld theory involved.) As a consequence one obtains a proof of 9.2.1 avoiding big Cohen{ Macaulay modules: if x1 . . . xn is a system of parameters of S , then there is a regular subring R of S^ in which x1 . . . xn generate the maximal ideal and over which S^ is nite (see A.22). Since the conclusion of 9.2.1 is invariant under completion, one obtains 9.2.1 in the same way as the implication `)' of 9.2.2. (b) In characteristic p the situation is just inverted: there is a direct proof of 9.2.1. Let be the maximal ideal of S . By 3.5.6, p
;!
n
6
H n (S ) = lim H n (xt) = 0: n
One has H n (xt ) = S=(xt), and the map S=(xt) ! S=(xt+i) is induced by the multiplication by xi1 xin. Since H n (S ) 6= 0, this map must be nonzero for t suciently large. Equivalently, xi1 xin 2= (xt1+i . . . xtn+i) for t suciently large and i t. On the other hand, if xt1 xtn 2 (xt1+1 . . . xtn+1), then one applies the Frobenius homomorphism repeatedly to obtain n
xtp1
e
xtpn 2 (xp1 +tp . . . xpn +tp ) e
e
e
e
e
which is a contradiction for e large. Via 9.2.2 this argument yields an `elementary' proof of 9.2.3. For still another proof of 9.2.1 in characteristic p see 176], as well as for a counterexample showing that normality is not sucient in characteristic p for R to have the direct summand property. 9.3 Canonical elements in local cohomology modules Independently of characteristic, the discussion in 9.2.4(b) shows that 9.2.1 and, hence, 9.2.3 are equivalent to the non-vanishing of certain elements in the local cohomology module H n (S ) (notation as in 9.2.1): one has xk1 xkn 2= (xk+1) for all k if and only if the image of 1 under n
358
9. Homological theorems
the map S ! S=(x) ! lim S=(xk ) = H n (S ) is non-zero. As the example ; ! S = K X ]] and x = X or x = X 2 shows, the element thus obtained depends heavily on the choice of the system of parameters for example, its annihilator varies with x. In the following we shall discuss a theorem which involves a `canonical element' in a local cohomology module although local cohomology does not appear explicitly: Theorem 9.3.1. Let (R k) be a Noetherian local ring of dimension n n
m
containing a eld. Let F. be a free resolution of the residue class eld k, and x a system of parameters. If : K. (x) F. is a complex homomorphism extending the natural epimorphism R=(x) k, then the homomorphism n : Kn (x) Fn is non-zero.
!
!
!
In order to derive a contradiction we assume that there exists a complex homomorphism with n = 0. There exists an x-regular module M by 8.4.2. Since (x) i for i large and M=xM 6= 0, we can pick an element y 2 M such that y 2= xM , but y 2 xM . The assignment 1 7! y then de nes a homomorphism R= ! M=xM . This homomorphism can be lifted to a complex homomorphism : F ! K (x M ) since the Koszul complex K (x M ) is acyclic see 1.6.12. Composition with gives a homomorphism = : K (x) ! K (x M ) with n = 0. The complex homomorphism extends the homomorphism 0 : R ! M with 0 (1) = y. As K (x M ) = K (x) M , one obtains a second such extension by 0 = idK (x) 0 . The complex K (x) is projective and K (x M ) is acyclic therefore and 0 di er only by a homotopy . In particular 0n = 0n ; n = n;1 @n: Proof.
m
m
m
.
.
.
.
.
.
@ ;;;;! Kn?(x) ;;;;! Kn;1(x) ? = y = 0 ;;;;! Kn (x M ): We may identify Kn (x) with R , Kn (x M ) = Kn (x) M with M and Kn;1(x) with R n. Then @n (R ) xR n, and so y = n0 (1) = n;1 @n(1) 2 xM , which
0
n
0
n;1
n
n
is a contradiction. Let us x the data x and F of the theorem. Complex homomorphisms and 0 both extending the epimorphism R=(x) ! k di er by a homotopy : .
@ ;;;;! Kn?(x) ;;;;! Kn;1 (x) ;;;;! ?? ? y y = = Fn;1 ;;;;! Fn ;;;;! ;;;;! Fn+1 ;;;;! ' '
0
n
n
n
n+1
0
n;1
n
n
359
9.3. Canonical elements in local cohomology modules
As above we identify Kn (x) with R furthermore we consider N = Ker 'n;1 = Im 'n as the target of 'n . (The module N is the n-th syzygy of k with respect to the resolution F .) Then 'n n(1) ; 'n n0 (1) = 'n n;1 @n (1): This element belongs to xN , since @n(1) xKn;1(x). So di erent choices of the complex homomorphism yield the same residue class ('n n (1)); 2 N=xN . On the other hand, given a complex homomorphism , we may freely choose to de ne 0 by 0 = + @ + ' . For the possible choices of n;1, the elements n;1 @n (1) exhaust xFn note that @n(1) = x1 e1 xnen with respect to a suitable basis of R n. In sum, n 6= 0 for every choice of if and only if 'n n (1) 2= xN for a speci c choice. Now consider the systems of parameters xt , t > 0. There is a natural map K (xt ) ! K (x) it sends ei ^ ^ eiu to xti ;1 xtiu;1ei ^ ^ eiu . Composition with : K (x) ! F gives a complex homomorphism t with nt (1) = xt1;1 xtn;1n (1). If all the homomorphisms : K (xt ) ! F which lift the epimorphism R=(xt) ! k have n 6= 0, then the arguments above imply xt1;1 xtn;1'n n(1) 2= xtN (1) for all t > 0: Observe that H n (N ) = ; lim N=xt N . So condition (1) is equivalent to the ! following: the image of 'n n (1) under the map N ! N=xN ! H n (N ) is non-zero. The module H n (N ) can also be represented as ; lim ExtnR (R= t N ) (see ! 3.5.3). Hence there is a natural homomorphism ExtnR (k N ) ! H n (N ). Moreover, the exact sequence 0 ! N ! Fn;1 ! ! F0 ! k ! 0 represents an element "(F ) 2 ExtnR (k N ) and thus an element !(F ) 2 H n (N ). The connection between `extensions' like the previous exact sequence and Extn is discussed in 264], pp. 82{87, or 48], Ch. X, x7 if one writes ExtnR (k N ) = HomR (N N )= (HomR (Fn;1 N )) where : N ! Fn;1 is the natural embedding, then "(F ) is the residue class of idN . The elements "(F ) and !(F ) may be called `canonical' since they depend functorially on F . In particular the vanishing of !(F ) is independent of F . Hochster 184] contains a detailed discussion of these facts and a proof of the following crucial statement: the element of H n (N ) constructed from a complex homomorphism : K (x) ! F as above can be identi ed with !(F ). The conclusion of 9.3.1 is therefore equivalent to !(F ) 6= 0, which justi es the name `canonical element theorem' for 9.3.1. As an application of 9.3.1 we prove a generalization of Krull's principal ideal theorem. (Another one will be given in 9.4.4.) Let M be a .
.
.
1
.
1
1
.
.
.
m
m
m
m
m
.
.
m
.
.
.
.
.
.
.
m
.
.
.
360
9. Homological theorems
module over a commutative ring R , and x 2 M . Then
O(x) = f (x): 2 HomR (M R)g
is called the order ideal of x. Every nitely generated ideal is an order ideal: given P x1 . . . xn 2 R, one sets M = R n and x = (x1 . . . xn). Obviously O(x) = Rxi. By Krull's principal ideal theorem O(x) has height n if R is Noetherian, provided O(x) is a proper ideal. For x 2 M = Rn this condition is equivalent to the existence of a maximal ideal such that x 2 M . The following theorem generalizes Krull's bound on height O(x) to arbitrary nite modules. In the general version the number n = rank R n must be replaced by m
m
big rank M = maxf(M ): p
p
2 Spec R minimalg:
If M has a rank, then big rank M = rank M . Theorem 9.3.2 (Eisenbud{Evans). Let (R k) be a Noetherian local ring containing a eld, and M a nite R-module. Then height O(x) big rank M m
for all elements x
2
m
M.
There is a prime ideal with height O(x) = height((O(x) + )= ). Let denote taking residue classes modulo . Every linear form M ! R ! R therefore O(x); O( x). Suppose the induces an R -linear form M . Then theorem has been proved for R and M big rank M: height O(x) = height O(x); height O( x) big rank M Proof.
p
p
p
p
if and only if x 2 M . Furthermore note that x 2 M As these arguments show, it suces to treat the case of an integral domain R . Then big rank M = rank M . Let h = height O(x) and n = dim R . There exists a system of parameters x1 . . . xn with x1 . . . xh 2 O(x). Replacing M by M Rn;h and x by x (xh+1 . . . xn), we may assume that O(x) is -primary. As usual, denotes HomR ( R ). Choose i 2 M such that a1 = 1(x) . . . an = n(x) is a system of parameters. The collection 1 . . . n de nes a map : M ! R n through (z ) = ( 1(z ) . . . n(z )). Let y = y1 . . . ym generate . Since x 2 M , there is a homomorphism : R m ! M with (y1 . . . ym ) = x. Let us put F = (R n) and G = (R m) , and de ne f : F ! G by f = . Then f `writes' a = a1 . . . an in terms of y i.e. f makes the diagram m
m
m
m
m
? ;;;;! R
F
?yf
G
a
;;;;! R y
361
9.4. Intersection theorems
commute. By 1.6.8 the exterior powers of f yield a complex homomorphism 0
;;! V?n F ;;! ;;! V?2 F ;;! ?F ;;! R ;;! 0 a
?yV f ?yV f ?yf y Vn+1 G ;;! Vn G ;;! ;;! V2 G ;;! G ;;! R ;;! 0 n
2
of Koszul complexes. By de nition f factors through M . So rank f V n rank M = rank M . On the other hand, f 6= 0 because there exists a complex homomorphism from K (y ) to a free resolution V F of k which extends the identity on k 9.3.1 guarantees that n n f 6= 0. Therefore rank f n and, hence, rank M n. Remarks 9.3.3. (a) Bruns 54] gave a more elementary proof of 9.3.2 which works for arbitrary local rings. (b) Formula (1) above shows that the canonical element theorem 9.3.1 implies 9.2.1 and thus the direct summand theorem 9.2.3. Surprisingly one can conversely derive 9.3.1 from 9.2.3 if the residue class eld of the local ring under consideration has characteristic p > 0 see Hochster 184]. (There seems to be no such derivation in characteristic zero.) Furthermore the main homological theorems like 9.4.3 and 9.5.6 can be derived from 9.3.1. (See 9.4.8 and 9.5.7.) (c) It is not dicult to reduce 9.3.1 to characteristic p via 8.4.1 see 184]. (Such a reduction will be carried out in detail for 9.4.3.) In connection with (b) and 9.2.4 that yields a proof of 9.3.1 which does not use big Cohen{Macaulay modules. .
.
9.4 Intersection theorems We have already met an intersection theorem in Section 8.2, the `new intersection theorem' (for local rings of characteristic p). We now want to prove a very powerful generalization and to derive several consequences, one of which will eventually explain why the results in this section are called `intersection theorems': it generalizes a variant of Serre's intersection theorem for the spectrum of a regular local ring. Theorem 9.4.1. Let (R k) be a Noetherian local ring containing a eld, and
m
' ;! ;! 2 2
' ;! ;! ;! ;!
F0 0 F. : 0 Fs Fs;1 F1 a complex of nite free R-modules such that codim F. 0. Let C = Coker '1 and e C, e = C. Then codim(Ann e) s. Proof. We use induction on dim(R=(Ann e)). Suppose dim(R=(Ann e)) = 0 rst. By 8.5.3 there exists a balanced big Cohen{Macaulay R -module M . s
m
1
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9. Homological theorems
Lemma 9.1.8 implies that F M is acyclic. One has depth M = dim R , and 9.1.2 yields depth(C M ) dim R ; s note that C M = Coker('1 M ). On the other hand, the natural surjection C M ! C= C M= M maps eM onto a module isomorphic to M= M . In particular, eM 6= 0. Since dim(R=(Ann e)) = 0, one has p (e M ) = 0 for some p, whence f g = Ass(e M) Ass C M. Therefore depth C M = 0, and so s dim R . Now suppose that dim(R=(Ann e)) > 0. There is nothing to prove if s = dim R . So we may assume that s < dim R . Let P be the nite set of prime ideals such that (i) Ann e and codim = codim(Ann e), or (ii) there exists i with Iri ('i) and codim = i. Then 2= P so we can choose x 2 such that x 2= for any 2 P . Let denote residue classes modulo x. It is a routine matter to verify that codim F 0. Furthermore e 2= C , and dim(R =(Ann e)) < dim(R=(Ann e)). The inductive hypothesis yields codim(Ann e) s. Since dim R dim R + 1, we have, as desired, codim(Ann e) = dim R ; dim(R=(Ann e)) dim R + 1 ; (dim(R =(Ann e)) + 1) s: The following corollary is usually called the `improved new intersection theorem': Corollary 9.4.2 (Evans{Grith). With the notation of 9.4.1 suppose that F R is acyclic for all 2 Spec R, 6= . If `(Re) < 1, then s dim R. Proof. Assume that s < dim R . In order to apply the theorem one must show that codim Iri ('i ) i for i = 1 . . . s. We even claim that height Iri ('i ) i. If this is false, then there exist j and a prime ideal Irj ('j ) with height = height Irj ('j ) < j . Since j s < dim R , one has 6= . On the other hand, the acyclicity of F R implies that grade(Irj ('j )) = grade Irj ('j R ) j by virtue of 1.4.13, which is a contradiction. Now that we can apply 9.4.1, we get a contradiction to our initial assumption s < dim R : `(Re) < 1 is equivalent with codim(Ann e) dim R . The next level of specialization is the `new intersection theorem' which for local rings of characteristic p was already proved in 8.2.6: Corollary 9.4.3 (Peskine{Szpiro, Roberts). With the notation of 9.4.1 suppose that F R is exact for all 2 Spec R, 6= . If s < dim R, .
m
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then the complex F. is exact. Proof. The complex F. satis es the hypothesis of 9.4.2, and furthermore `(Re) < for all e Coker '1. So Coker '1 = 0 by Nakayama's lemma. The map '1 is a split epimorphism, and we obtain a shorter complex which also satis es the hypothesis of the corollary. Induction on s yields
1
the assertion.
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At least once in this chapter we want to give a complete proof of a theorem by direct reduction to characteristic p via Hochster's niteness theorem 8.4.1. Since we have a proof of 9.4.3 in characteristic p which is independent of big Cohen{Macaulay modules, 9.4.3 is the best candidate for such a demonstration. Second proof of 9.4.3. Suppose that 9.4.3 is violated for a local ring containing a eld of characteristic zero. Arguing as in the proof of 8.2.6, we may assume that F0 6= 0 and Im '1 F0. Choose a basis for Fi , i = 0 . . . s. Then each homomorphism 'i is represented by a matrix Ai = (a(jli)). Since F is a complex, one has m
.
(2)
Ai;1Ai = 0
i = 1 . . . s:
Let x be a system of parameters for R . That Im '1 F0 , can be expressed by the relation (ajl(i))t 2 xR (3) for some t > 0 and all i j l . That F R is exact for all 6= , is described by the following two conditions: (i) F R is split acyclic for each 2 Spec R , 6= P (ii) si=0(;1)i rank Fi = 0. Let ri be the expected rank of 'i . Via 1.4.12 condition (i) can be translated into the non-vanishing of Iri (Ai) modulo for all prime ideals 6= , equivalently (4) (xR )u Iri (Ai) for some u > 0 and all i = 1 . . . s. It is mechanical to express (2) { (4) in terms of polynomial equations over Z satis ed by the entries of the matrices, the elements of x, and the coecients in the linear combinations involved. These equations only depend on the numerical parameters s, rank Fi , t, and u. Conversely, given a solution to one of the systems of equations thus obtained, one immediately constructs a counterexample to 9.4.3, interpreting the matrices as homomorphisms. The reader is invited to try similar reductions for 9.4.1, 9.5.2, and 9.5.5. The next member of the chain of corollaries is the `homological height theorem'. It belongs to the class of `superheight theorems'. For a proper ideal I of a Noetherian ring R let us de ne its superheight as the supremum of height IS where S is any Noetherian ring to which there exists a ring homomorphism R ! S , with IS 6= S . The fundamental superheight theorem is Krull's principal ideal theorem it says that superheight I is bounded above by the minimal number of generators of I . m
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Theorem 9.4.4 (Hochster). Let R be a Noetherian ring containing a eld, and M 6= 0 a nite R-module. Then superheightAnn M proj dim M. Before proving this theorem one should note that it is a far-reaching generalization of Krull's principal ideal theorem for Noetherian rings containing a eld k: take R = kX1 . . . Xn] and M = k = R=(X1 . . . Xn). Then proj dim M = n, and therefore height(x1 . . . xn ) n for elements x1 . . . xn of a K -algebra S with (x1 . . . xn) 6= S . (Simply consider the extension R ! S induced by the substitution Xi ! xi .) Proof of 9.4.4. The theorem is trivial if proj dim M = 1, so assume it is nite, and let R ! S be a Noetherian extension of R such that (Ann M )S 6= S . Replacing S by a localization S for a minimal prime ideal of (Ann M )S and R by R \R , one may assume that R ! S is a local extension, and (Ann M )S is not contained in any prime ideal of S di erent from the maximal ideal of S . Let F be a minimal free resolution of M over R . Then \ R 6 Ann M for every 2 Spec S with 6= . Hence M R \R = 0, and F R \R is split exact. Split exactness is preserved under ring extensions, and so F S is split exact: F S satis es the hypotheses of 9.4.3, whence proj dim M dim S . Let k be an algebraically closed eld, and Y Z subvarieties of the ane space An (k) (or the projective space Pn (k)). Then a classical theorem of algebraic geometry asserts that dim W dim Y + dim Z ; n for every irreducible component of Y \ Z (152], Prop. 7.1). If are the prime ideals de ning the varieties Y Z , and W respectively, then this inequality can be written (5) height height + height : Note that is a minimal prime ideal of + . Serre showed in 334], Theor!eme 3, p. V-18 that the inequality (5) holds for prime ideals of a regular local ring such that is a minimal prime ideal of + . Suppose that = is the maximal ideal of R . Then contains all the minimal prime ideals of any ideal I R , and we can replace and by arbitrary ideals I and J to obtain the following version of Serre's theorem: let I J be ideals of a regular local ring (R ) such that I + J is -primary then height I + height J dim R , or, returning to dimensions, (6) dim R=I dim R ; dim R=J: The example R = kX1 X2 Y1 Y2 ]]=(X1Y2 ; X2Y1 ), I = (x1 x2 ), J = (y1 y2), shows that the last inequality is false in non-regular local rings. However, one can hope that in the presence of their characteristic property q
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9.4. Intersection theorems
namely nite projective dimension of nite modules, one can generalize the inequality above, reading I and J as the annihilators of modules M and N . The best possible result to be expected is the direct generalization of (6): (7) dim N dim R ; dim M for all modules M N over a local ring (R ) such that M has nite projective dimension and Supp M \ Supp N = f g. It seems to be unknown whether (7) holds, but (7) turns into a valid inequality if we replace its right side by depth R ; depth M = proj dim M (the Auslander{ Buchsbaum formula see 1.3.3). It should now be clear why the following corollary is named the `intersection theorem'. (It is customary in this context to express the condition Supp M \ Supp N = f g by `(M N ) < 1, which is an equivalent requirement if M N 6= 0.) Theorem 9.4.5 (Peskine{Szpiro). Let R be a Noetherian local ring containing a eld, and M N 6= 0 nite R-modules such that `(M N ) < 1. Then dim N proj dim M. Proof. There is nothing to prove if proj dim M = 1. So assume it is nite. Neither the condition `(M N ) < 1, nor the number dim N , can change if we replace N by another nite module with the same support. In particular we may replace N by S = R= Ann N . Then (Ann M )S is primary to the maximal ideal of S , and the desired inequality proves to be a special case of 9.4.4. It is easy to generalize 9.4.5 to situations in which `(M N ) is not necessarily nite. Suppose that dim(M N ) > 0. Then dim N > 0, and none of the nitely many minimal prime ideals of M N or N equals . Therefore there exists x 2 such that dim N=xN = dim N ; 1 and dim(M (N=xN )) = dim(M N ) ; 1. Applied inductively, this argument proves the following corollary: Corollary 9.4.6. Let R be a Noetherian local ring containing a eld, and M N 6= 0 nite R-modules. Then dim N proj dim M + dim(M N ). One of the reasons for which we have stated the corollary, is that it explains why 9.4.5 is easier to prove than inequality (7) above: 9.4.6 is equivalent to dim R ; dim M depth R ; depth M for N = R . The following theorem, often called `Auslander's conjecture', does not strictly fall under the title of this section, but its proof is short and an elegant application of the intersection theorem 9.4.5. Theorem 9.4.7 (Peskine{Szpiro). Let (R ) be a Noetherian local ring containing a eld, and M 6= 0 a nite module of nite projective dimension. m
m
m
m
m
m
Then every M-sequence is an R-sequence in particular every M-regular element is R-regular.
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9. Homological theorems
If x 2 R is regular on M and on R , then proj dimR=(x) M=xM = proj dim M by 1.3.5. Thus it is enough to prove the second statement the rst follows by induction. One has to show that every 2 Ass R is contained in some 2 Ass M . We proceed by induction on dim M . If dim M = 0, then 2 Ass M , and certainly . Assume that dim M > 0. If there is a prime ideal 2 Supp M such that 6= , one can apply the inductive hypothesis to M : there exists 0 2 Spec R with 0 2 Ass M and 0 R hence 0 \ R satis es our needs. Otherwise V ( ) \ Supp M = f g. So dim R= proj dim M by 9.4.5. On the other hand depth R dim R= according to 1.2.13, and furthermore the Auslander{Buchsbaum formula says that proj dim M + depth M = depth R . Therefore depth M = 0, whence 2 Ass M . Remarks 9.4.8. (a) The new intersection theorem 9.4.3 was proved for all local rings by Roberts 314]. Consequently 9.4.4{9.4.7 and 9.4.9{ 9.4.10 hold without the hypothesis that R contains a eld. In particular 9.4.4 is a true generalization of Krull's principal ideal theorem (take R = ZX1 . . . Xn]). (b) It is possible to avoid the use of big Cohen{Macaulay modules in the proof of the improved new intersection theorem 9.4.2. In fact, 9.4.2 is on a par with the canonical element theorem 9.3.1. Hochster 184] derived 9.4.2 from 9.3.1, and Dutta 82] found the converse. As pointed out in 9.3.3, the canonical element theorem can be proved independently of the existence of big Cohen{Macaulay modules. (c) The intersection theorem 9.4.5 can be improved to the best conceivable result if M is perfect see 297], p. 94, Theor!eme 4.2: (i) grade M + dim M = dim R (ii) if N is a nite R -module such that l (M N ) < 1, then dim M + dim N dim R . L Furthermore both (i) and (ii) hold if R = 1i=0 Ri is a graded ring with R0 an Artinian local ring, M is a nite graded R -module of nite projective dimension, and N is a nite graded R -module see Peskine and Szpiro 299]. Equation (i), sometimes called the `codimension conjecture', was proved by Foxby 116] for modules M of nite projective dimension over a large class of equicharacteristic local rings. (d) Let R be a Noetherian ring, and M N nite R -modules such that proj dim M < 1 and `(M N ) < 1. Then the modules TorRi (M N ) have nite length, and only nitely many are non-zero. Thus one can de ne the intersection multiplicity of M and N by Proof.
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e(M N ) =
1 X i=0
(;1)i `(TorRi (M N )):
This notion was introduced by Serre 334]. He proved that the following
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9.5. Ranks of syzygies
hold if (R ) is an unrami ed regular local ring (see 270] for this notion): (i) if dim M + dim N < dim R , then e(M N ) = 0 (ii) if dim M + dim N = dim R , then e(M N ) > 0. (Note that dim M +dim N dim R , as discussed above.) Recently Gabber 121] showed that over an arbitrary regular local ring one has (M N ) 0. However, both (i) and (ii) fail if R is allowed to be an arbitrary local ring: Dutta, Hochster, and McLaughlin 85] constructed counterexamples over the hypersurface ring kX1 X2 Y1 Y2 ]=(X1Y2 ; X2Y1 ). However, (i) was shown to hold if both M and N have nite projective dimension and R is a complete intersection (Roberts 313], 315], Gillet and Soule 125]) or dim Sing R 1 (315]). m
Exercises 9.4.9. A Noetherian local ring (R
) containing a eld is Cohen{Macaulay if (and only if) there exists an R -module of nite length and nite projective dimension. Prove this. 9.4.10. Let ' : R ! S be a surjective homomorphism of Noetherian local rings containing a eld such that proj dimR S < 1. Show the following are equivalent: (i) R is Cohen{Macaulay and S is a perfect R -module (of type 1) (ii) S is Cohen{Macaulay (Gorenstein). Hint: 9.4.9 is essential for the di cult implication (ii) ) (i). 9.4.11. Prove the assertions on perfect R -modules in 9.4.8(c) for Noetherian local rings R containing a eld. Hint: It su ces to prove that grade M + dim M dim R which is quite evident. 9.4.12. Let R be a Cohen{Macaulay local ring, and x a system of parameters for R . Show that e(x N ) = e(R=(x) N ) for all nite R -modules N . m
9.5 Ranks of syzygies Let R be a local ring, and M a nite R -module of nite projective dimension. Then proj dim M depth R : the length of a minimal free resolution is bounded by depth R . Moreover, each of the values s = 0 . . . depth R occurs if we choose M = R=(x) with an R -sequence x = x1 . . . xs . In this section we shall discuss the possible values for the Betti numbers of M and the ranks of its syzygy modules. For systematic reasons and in view of an application to Bass numbers below, it is useful to consider a larger class of complexes than just minimal free resolutions, namely minimal complexes of codimension 0. Let M be a module over a commutative ring R , and x 2 M . The notion of order ideal, which was introduced in connection with 9.3.2, plays an important role in the following. The next lemma describes a property of x which is controlled by O(x).
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Lemma 9.5.1. Let R be a Noetherian ring, M a nite R-module, x 2 M,
and a prime ideal. Then x generates a non-zero free direct summand of M if and only if (x). Proof. Since HomR (M R ) is naturally isomorphic to Hom(M R ) , the
6 O
p
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p
formation of order ideals commutes with localization. We may therefore assume that (R ) is a local ring. If M = Rx N and Rx = R , then there obviously exists 2 HomR (M R ) such that (x) = 1. Conversely, if (x) = 1, then M = Rx Ker . Suppose now that ' : F ! G is a map of nite free modules. Let e 2 F . Given a basis g1 . . . gn of G, there are uniquely determined elements a1 . . . an 2 R such that '(e) = a1g1 + + angn . The elements g1 . . . gn of the dual basis of HomR (G R ) yield the values gj('(e)) = aj . Therefore OG ('(e)) = (a1 . . . an ). Theorem 9.5.2. Let (R ) be a local ring containing a eld, and p
p
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m
' ;! ;!
' ;! ;! ;! ;! O ;
Fs Fs;1 F1 F0 0 F. : 0 a complex of nite free R-modules. Then, for j = 1 . . . s and every e Fj with e = Fj + Im 'j +1, one has codim ('j (e)) codim F. + j. Proof. Let t = codim F. . For given j we truncate the complex at Fj ;1, and adjust the indices by setting Fi0 = Fi+j ;1 and t0 = t + j 1. Replacing the given data by those just de ned, we may assume that j = 1. Let J = ('1(e)). There is something to prove only if J and codim F. 0. We put R = R=J and F = F. R . From the description of J preceding the theorem one sees that ' 1( e) = 0. In order to derive a contradiction, we assume that codim J t. Note that Iri ( 'i) = (Iri ('i ) + J )=J . Hence s
2
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dim(R =Iri ( 'i)) dim(R=Iri ('i )) dim R ; i ; t dim R ; i: This inequality shows that codim F 0. Let M be a balanced big Cohen{Macaulay module for R . By virtue of 9.1.8, F M is acyclic. Since ' 1( e) = 0, we have ( '1 M )( e M ) = 0. Let C = Coker '2, and : F1 ! C be the natural epimorphism. Since F M is acyclic, ' 1 M induces an isomorphism C M ! Im( '1 M ). So ( e) M = 0. On the other hand, the hypothesis e 2= F1 + Im '2 implies that ( e) 2= C . Thus the image of ( e) M under the natural epimorphism C M ! (C = C ) (M= M ) is isomorphic to M= M 6= 0. This is a contradiction. An application of the following corollary was anticipated in the proof of 2.3.14. Corollary 9.5.3. Let (R k) be a regular local ring containing a eld, and .
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I an ideal generated by a sequence x. Then the natural homomorphism from Hi (x k) = K. (x) k to TorRi (R=I k) is zero for i > grade I. m
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9.5. Ranks of syzygies
The natural homomorphism Hi (x k) ! TorRi (R=I k) is induced by a complex homomorphism from K (x) to a free resolution F of R=I see 1.6.9. It only depends on I and x, so that we may assume that F is a minimal free resolution. Since R is regular, F has nite length by 2.2.7. That H (x k) = K (x) k and TorR (R=I k) = F k, follows from the minimality of the complexes K (x) and F . Thus the map H (x k) ! TorR (R=I k) is just k. The assertion amounts to (Ki (x)) Fi for i > grade I . Let z 2 Ki (x), and @ and ' denote di erentiation in K (x) and F . If (z ) 2= Fi , then grade O((@(z ))) = codim O('((z ))) i by 9.5.2: an acyclic complex has non-negative codimension as observed above. On the other hand, O('((z ))) = O((@(z ))) I since Im @ IK (x). As indicated above, we aim at a bound for the expected ranks ri of the maps in a free complex F . Reasoning inductively, we will have to pass to a complex 0 ! Fs ! Fs;1 ! ! F2 ! F10 ! F00 ! 0 in which rank F10 = rank F1 ; 1. Theorem 9.5.2 enables us to nd F10 , whereas the following lemma contains the construction of F00 . Lemma 9.5.4. Let R be a Noetherian ring and M a nite R-module. Then there is a nite free R-module F and a homomorphism ' : M ! F with Proof.
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the following property: If is a prime ideal, and N M is a free direct R -summand of rank r, then (' R )(N ) is a free direct R -summand of F with rank(' R )(N ) = r. p
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Let denote the functor HomR ( R ). There is a nite free R module G with an epimorphism : G ! M . Let h : M ! M be the canonical homomorphism, and choose ' = h, F = G. Then ' : M ! F has the property that every linear form 2 M can be extended to F along '. Since R is Noetherian and the modules involved are nite, the preceding construction commutes with every localization of R . Thus assume R = R . Now the hypothesis on N is equivalent to the existence of g1 . . . gr 2 N and 1 . . . r 2 M such that N = Rg1 + + Rgr and i (gj ) = ij . Since the i can be extended to F , the elements '(g1) . . . '(gr ) generate a free direct summand of rank r. As pointed out before 9.1.9 every complex of nite free modules over a local ring decomposes into a split exact direct summand and a direct summand which is minimal. For the ranks of the maps in a split exact complex one can only say that they are non-negative, but for those of a minimal complex there exists a non-trivial lower bound. (It was essentially given by Evans and Grith in the form of Corollary 9.5.6.) Proof.
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Theorem 9.5.5. Let (R ) be a local ring containing a eld, and m
' ;! ;!
' ;! ;! ;! ;!
F. : 0 Fs Fs;1 F1 F0 0 a length s minimal complex of nite free R-modules. Let ri denote the expected rank of 'i. If codim F. 0, then ri codim F. + i for i = 1 . . . s 1. s
1
;
The same manipulation as in the proof of 9.5.2 reduces the theorem to a statement about r1 . Since the theorem makes an assertion only on r1 . . . rs;1, the complex which remains after the truncation has length 2 there is nothing to prove if s = 1. We introduce an auxiliary variable t, and use induction on t to show that codim F t implies r1 t + 1. Since codim F 0, Lemma 9.1.8 yields acyclicity of F M for a balanced big Cohen{Macaulay module M of R such a module exists by 8.5.3. Therefore 9.1.9 implies ri 1 for i = 1 . . . s. This inequality covers the case t = 0, and shows furthermore that F1 6= 0: one has rank F1 = r1 + r2 2. So there exists e 2 F1 with e 2= F1 . Since F is minimal, e 2= F1 + Im '2. Let t 1. Put F10 = F1 =Re, and choose '02 as the induced map F2 ! F10 . Applying 9.5.4 to Coker '02 one obtains a homomorphism Coker '02 ! F = F00 . Its composition with the natural epimorphism F10 ! Coker '02 then yields '01 : F10 ! F00 . For the complex Proof.
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;! ;!
' ' ;! ;! ;! ;! ;! ; ; 6 particular we have a decomposition (F1 ) = (Im '2 ) (Coker '2) with rank(Im '2) = r2 and rank(Coker '2 ) = r1 . Moreover { and this is the crucial argument { codim O('1(e)) t + 1 by 9.5.2. Therefore '1(e) 0
0
F.0 : 0 Fs Fs;1 F2 F10 F00 0 0 0 0 one has r2 = r2, r1 = r1 1. In order to show that codim F. t 1 we must verify the following inequalities: (i) codim Ir0 ('02) t + 1, and (ii) codim Ir0 ('01 ) t. For (i) we choose a prime ideal with codim t. Certainly Iri ('i ) for i = 1 . . . s. Therefore F. R is split acyclic by 1.4.12. In 2
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generates a free direct summand of (F0 ) by 9.5.1. A fortiori the residue class e of e generates a non-zero free direct summand of (Coker '2 ) . So (Coker '02) = (Coker '2 ) =R e is free of rank r10 = r1 ; 1, and the exact sequence 0 ;! (Im '02) ;! (F10 ) ;! (Coker '02 ) ;! 0 splits. Also this shows that (Im '02) is a free direct summand of rank r2 = r20 of (F10 ) . By 1.4.8 we get Ir0 ('02) 6 . Since is an arbitrary prime ideal with codim t, the inequality (i) has been proved. p
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9.5. Ranks of syzygies
Slightly more than required for (ii) we show that codim(R=Ir0 ('01)) t + 1. Pick as before. We saw that (Coker '02) is free of rank r10 . Since '01 was constructed as prescribed by 9.5.4, (Coker '02) is mapped isomorphically onto a free direct summand of F00 . As desired, Ir0 ('01) 6 . If it should happen that F 0 is not minimal, then one splits o a direct summand id: R u ! R u from F10 ! F00 . This does not a ect the codimension, and even improves the desired inequality r10 t which holds by induction. (Because of s 2 the construction of F 0 does not touch Fs , so that F 0 also has length s.) Corollary 9.5.6 (Evans{Grith). Let R be a Noetherian local ring containing a eld, and M 6= 0 a module of projective dimension s < 1. Then (a) for i = 1 . . . s ; 1 the i-th syzygy Mi of M has rank i, (b) ( 2i + 1 i = 0 . . . s ; 2, i = s ; 1, i (M ) s 1 i = s. 1
p
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A minimal free resolution F of M is acyclic, and thus has codimension 0, as was observed above. Theorem 9.5.5 says that for i = 1 . . . s ; 1 the i-th map 'i has expected rank ri i. Since F is acyclic, ri = rank 'i = rank Mi see 1.4.6. This proves (a) from which (b) follows with i (M ) = ri + ri+1. (Note that s (M ) > 0 because of proj dim M = s.) It is of course not dicult to give a non-local version of the corollary, which we leave to the reader. Remarks 9.5.7. (a) Corollary 9.5.6 is the best possible result. In fact, if R is a Noetherian local ring, and M the m-th syzygy module of a module of nite projective dimension, then M contains a free submodule L such that M=L inherits this property and rank M=L m see Bruns 53]. Similarly one can nd modules M for all preassigned values of proj dim M = s depth R and i(M ), i = 0 . . . s, which are consistent with 9.5.6. (b) It is not necessary to use big Cohen{Macaulay modules in the proof of 9.5.6. Ogoma 295] derived it from the improved new intersection theorem 9.4.2. (c) Theorem 9.5.2 and its consequences admit conclusions even for local rings not containing a eld. Let p = char R= . Then one passes from a given complex F over R to F R=(p), and R=(p) contains a eld. The reader may verify that codim O('j (e)) codim F + j ; 1 in 9.5.2, regardless of whether dim R=(p) = dim R , or dim R=(p) = dim R ; 1. Similarly the bounds in 9.5.3, 9.5.5, and 9.5.6(a) become worse by at most 1. Proof.
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9. Homological theorems
9.6 Bass numbers Let R be a Noetherian ring, and M a nite R -module. The Bass numbers i ( M ) = dimk( ) ExtiR (k( ) M ) 2 Spec R determine the modules in a minimal injective resolution I : 0 ;! E 0 (M ) ;! E 1(M ) ;! ;! E i (M ) ;! L of M by 3.2.9 one has E i (M ) = 2Spec R E (R= )i ( M ) for all i 0. In this section we want to derive inequalities satis ed by the numbers i( M ) when (R ) is a local ring since the Bass numbers are local data by de nition, such inequalities can be translated into assertions about the i ( M ) in general. ^ ^ k). Suppose that (R k) is a local ring with -adic completion (R i i i Since ExtR (k M ) = ExtR (k M ) R^ = ExtR^ (k M^ ) for all i 0, one ^ has i ( M ) = i ( ^ M ). Therefore it is no restriction to assume R is complete. For simplicity of notation we set i = i( M ). By their very de nition the local cohomology modules of M are given as H i (M ) = H i (; (I )), see Section 3.4. It is easy to determine ; (I ) since a non-zero element of E (R= ) cannot be annihilated by a power of if 6= see 3.2.7. Thus ; (I ) is the subcomplex p
p
p
p
p
p
.
p
p
p
m
m
p
m
m
m
m
m
m
.
.
m
m
m
p
.
m
p
m
m
;! E(k) ;! ;;! E (k) ;! By Grothendieck's theorem 3.5.7 we have H i (M ) = 6 0 for i = depth M and i = dim M , in particular i = 6 0 for these values of i. On the other
J. : 0
i;1
0
0
i
i
m
hand, i = 0 for i < depth M . By assumption R is complete. So Theorem 3.2.13 yields HomR (E (k) E (k)) = R and one obtains a complex of nite free modules from an application of the functor HomR (E (k) ) to J : .
G. = HomR (E (k) J . ): 0
;! R ;! R ;! ;;! R ;! 0
0
i;1
1
i
i
Moreover there is some information on the maps i and i. The endomorphisms of E (k) are just given by multiplication by elements of R therefore the maps i can naturally be considered as matrices over R , and i is given by the same matrix as i. Since I is a minimal injective resolution, the entries of these matrices are in . Also, one obtains a complex of nite free R -modules if one applies HomR ( E (k)) to J : .
m
.
L. = HomR (J . E (k)):
R ;;! ;! R ;! R ;! 0 ;! i
i
i;1
1
1
0
0
373
9.6. Bass numbers
the matrix representing i is the transpose of i . Let denote the functor HomR ( R ). As just seen, (G ) = L (L ) = G : The advantage of L over G is that we know its homology. By the exactness of HomR ( E (k)), Hi (L ) = HomR (H i (J ) E (k)) = HomR (H i (M ) E (k)): We ;claim that dim Hi(L ) i: for this to hold it is surely sucient that dim R=(Ann H i (M )) i, and the latter inequality has already been proved in 8.1.1. In order to adapt the present notation to that in the previous section we set d = dim R )i = d ;i 'i = d ;i and de ne the complex F by .
.
.
.
.
.
.
.
m
.
m
.
' ;! R ;! R ;! ;! R ;! R ;! 0: We want to show that codim F 0. We consider the truncation (L jd ; i + 1): R ;;! R ;! ;! R ;! R ;! 0: Since dim Hv (L ) v, the complex (L jd ; i + 1) R is exact, and thus split exact for prime ideals satisfying codim i ; 1. We dualize to get that ' 0 ;! R ;! R ;! ;! R ;! 0 is split acyclic. Thus 1.4.12 gives Ir ('i) 6 , and codim Ir ('i ) i as desired. Let t = depth M . As noticed above, R = 0 for j 1, R = 6 0, and F is a minimal complex of length d ; t. Now we have reached our goal 9.5.5 yields ( 1 i = d ; t, )i = ri+1 + ri d ; t i = d ; t ; 1, 2i + 1 i = 0 . . . d ; t ; 2.
F. : 0
'd
d
d ;1
1
1
0
.
d ;i
d ;i+1
.
d ;i
.
.
0
p
p
d
0
1
p
d
d ;1
i;1
i
p
d ;t+j
i
d ;t
.
Returning to the previous notation we get part (a) of Theorem 9.6.1. Let R be a Noetherian local ring containing a eld, dim R = d, and M a nite R-module of depth t.
(a) Then
i = t,
; t i = t + 1, 2(d ; i) + 1 i = t + 2 . . . d. (b) If t < dim M = d, then d ( M ) 2. i ( M ) m
( 1 d
m
374
9. Homological theorems
(a) was proved above. In its proof we exploited results on the vanishing of local cohomology and its non-vanishing at the depth of a module. Part (b) relies on its non-vanishing at the dimension, as will be seen now. d d ; Consider the interval R d ;! R d ;;! R d ; of L . Its homology at R d is Hd (L ) = HomR (H d (M ) E (k)), and the transpose of d ;1 is the map '1 in Proof.
1
+1
.
F. : 0
1
.
m
' ;! R ;! ;! R ;! R ;! 0: d
1
1
0
Suppose that )0 = d = 1. Since depth M < d , the arguments that proved (a) yield that r1 1 (with respect to F ), hence r1 = 1 note that r1 )0 . Furthermore dim(R=Ir ('1)) d ; 1, as stated above. This implies '1 R is surjective for prime ideals with dim R= = dim R . Therefore d ;1 R is injective, and dim Hd (L ) < d . We choose a Gorenstein ring S with an epimorphism S ! R . By the variant 3.5.14 of the local duality theorem .
1
p
p
p
p
.
Hd (L. ) = HomR (H d (M ) E (k)) = ExtnS;d (M S ) m
n = dim S:
Let 2 SuppS M with dim S= = d . Then ExtnS;d (M S ) = 0, since dim Hd (L ) < d that however contradicts 3.5.11 (note that dim M = 0, dim S = n ; d ). Two corollaries are immediate. The rst of them is usually called `Bass' conjecture' the second was conjectured by Vasconcelos. Corollary 9.6.2 (Peskine{Szpiro). Let R be a Noetherian local ring containing a eld. If R has a nite module M 6= 0 of nite injective dimension, q
q
q
q
q
.
q
q
then R is a Cohen{Macaulay ring.
In fact, if inj dim M is nite, then it equals depth R by 3.1.17. The theorem yields inj dim M dim R . The converse could already have been proved in Chapter 3. Let (R k) be a local Cohen{Macaulay ring, x a system of parameters, and E the injective hull of k over R . Then HomR (R=(x) E ) has nite length by 3.2.12. The Koszul complex K (x) is a projective resolution of R=(x). Therefore the acyclic complex K (x E ) = HomR (K (x) E ) is an injective resolution of K 0 (x E ) = HomR (R=(x) E ). Corollary 9.6.3 (Foxby). Let R be a Noetherian local ring containing a eld, and d = dim R. If d ( R ) = 1, then R is a Cohen{Macaulay ring, m
.
.
.
hence Gorenstein.
m
Remarks 9.6.4. (a) Both the corollaries hold for all local rings: (i) Roberts 312] gave a characteristic-free proof of 9.6.3. It exploits the properties of dualizing complexes. Kawasaki 233] generalized 9.6.3 using the methods of this section: a complete local ring of type n
375
9.6. Bass numbers
satisfying Serre's condition (Sn;1) is Cohen{Macaulay (for n = 2 one has additionally to assume that R is unmixed). (ii) For a large class of local rings, 9.6.2 was rst proved by Peskine and Szpiro. Their argument rests mainly on the intersection theorem 9.4.5 and the following fact which is interesting in itself: let (R ) be a Noetherian complete local ring, and M 6= 0 a nite R-module of m
nite injective dimension then there exists a nite R-module N such that
proj dim N = depth R ; depth M and Supp N = Supp M. Since Roberts 314] proved the intersection theorem for all local rings, 9.6.2 holds without any restriction. The theorem of Peskine{Szpiro just mentioned can be proved by the method we used for 9.6.1, independently of the hypothesis that R contains a eld. (One constructs the complex F as in the proof of 9.6.1 and chooses N = Coker 'd ;u+1 where d = dim R , u = depth R = inj dim M .) On the other hand, it can also be obtained as a consequence of 9.6.2 in conjunction with Exercise 9.6.5. In fact, if R contains a eld and has a nite module of nite injective dimension, then it is Cohen{Macaulay by 9.6.2. Furthermore it has a canonical module since it is complete, and thus it satis es the hypothesis of 9.6.5. (b) Using 9.5.7(c) one can derive slightly weaker bounds for Bass numbers over an arbitrary Noetherian local ring. (c) If R is a Cohen{Macaulay ring, then the complex F above is acyclic, and already 9.1.9 gives .
.
i ( M ) m
1
i = depth M and i = dim R , 2 depth M < i < dim R .
This inequality and 9.6.1(b) were rst obtained by Foxby 115] for Cohen{ Macaulay rings and local rings containing a eld. (d) Whenever d ( M ) > 0, d = dim R , and inj dim M = 1, then i( M ) > 0 for all i dim R see 3.5.12. m
m
Exercise 9.6.5. Let R be a Cohen{Macaulay local ring with canonical module !. Recall from Exercise 3.3.28 that a nite R -module of nite injective dimension has a minimal augmented !-resolution . : 0 ! !rp ! ! !r ! M ! 0 with p = dim R ; depth M . The following assertions (due to Sharp 339]) set up a bijective correspondence between nite modules M of nite injective dimension and those of nite projective dimension that is given by the assignment M 7! HomR (! M ) and its inverse N 7! N !. (a) Let N be a nite R -module of nite projective dimension with minimal free resolution F. . Show that F. ! is a minimal !-resolution of M = N ! in particular dim R ; depth M = proj dim N and Supp M = Supp N . 0
376
9. Homological theorems
(b) Conversely, let M be a nite R -module of nite injective dimension with minimal !-resolution . . Show that HomR (! . ) is a minimal free resolution of N = HomR (! M ). (c) Using 9.5.7(a) show that 9.6.1 gives the best possible lower bounds for the Bass numbers of an R -module. Hint: use 9.1.6 for (a) and (b), noting that ! is a Cohen{Macaulay module with Supp ! = Spec R and that End(!) = R .
Notes The acyclicity criterion 9.1.6 is essentially due to Buchsbaum and Eisenbud 63]. The general version without any niteness condition on the ring R or the module M was given by Northcott 290]. The concept of grade on which it is based goes back to Hochster 179]. To us it seemed most convenient to use Koszul homology in the de nition of grade. Section 9.2 is based on Hochster's article 176]. We outlined the fact that essentially all the homological theorems can be derived from the direct summand theorem 9.2.3 or its equivalent, the monomial theorem 9.2.1. One of the rare results in mixed characteristic is due to Hochster and McLaughlin 199] it says that a regular local ring is a direct summand of a nite extension domain if the extension of the elds of fractions has degree two. As a surprising spin-o of an investigation of the monomial theorem in mixed characteristic, Roberts 317] obtained a counterexample for Hilbert's fourteenth problem, and furthermore a prime ideal in a formal power series ring whose symbolic Rees algebra is not nitely generated. The material on the canonical element theorem in Section 9.3 is taken from Hochster's comprehensive treatise 184]. It seems however that the idea to compare a Koszul complex for a system of parameters with a free resolution of the residue class eld, was rst used by Eisenbud and Evans 92] in the demonstration of their generalized principal ideal theorem 9.3.2. Hochster 184] contains many more results than indicated in 9.3.3 and 9.4.8. In particular we would like to mention a connection between canonical elements and canonical modules. The canonical element theorem has also been studied by Dutta 82], 84], and Huneke and Koh 216]. A `tremendous breakthrough' (Hochster 186], p. 496) was made by Peskine and Szpiro in 297]. As mentioned already in Chapter 8, they were the rst to apply the Frobenius morphism in the context of homological questions and to reduce such questions from characteristic zero to characteristic p through Artin approximation. They proved the intersection theorem 9.4.5 in characteristic p and for local rings which can be obtained as inductive limits of local e tale extensions of localizations of ane algebras over a eld of characteristic zero. Furthermore, for the
Notes
377
same class of local rings they were able to deduce Auslander's conjecture 9.4.7 and Bass' conjecture 9.6.2 from the intersection theorem. An equally fundamental achievement is Hochster's construction of big Cohen{Macaulay modules. It enabled him to extend Peskine and Szpiro's results to all local rings containing a eld, and had the side-e ect of a considerable technical simpli cation. See 175], 178], 181]. The new intersection theorem is due independently to Peskine and Szpiro 299] and Roberts 309]. It seems that Foxby 115] published the rst complete proof valid for all equicharacteristic local rings using big Cohen{Macaulay modules he gave an even more general theorem than 9.4.3. As pointed out above, Roberts 314], 316] proved the new intersection theorem in full generality it has been noted which of the theorems therefore become valid without a restriction on the characteristic. The improved new intersection theorem 9.4.2 is implicitly contained in Evans and Grith 97] it was explicitly formulated (and given its name) by Hochster 184]. Still another extension of the intersection theorem must be mentioned, namely Foxby's version for complexes in 116]. In 9.4.8 we commented on generalizations of Serre's theorem for intersection multiplicities. It should be added here that some positive results were obtained by Foxby 117] and Dutta 80], 81]. The original argument of Evans and Grith's remarkable syzygy theorem 9.5.6 is found in 97]. It requires a weak condition on the underlying ring. Such conditions were removed by Ogoma 295], as pointed out in 9.5.7. Our proof of the more general result 9.5.5 is a direct generalization of the argument in 98]. This monograph of Evans and Grith contains an extensive discussion of questions related to the syzygy theorem its bibliography gives an overview of the pertinent literature. Successively better inequalities for Bass numbers were obtained by Foxby 112], Fossum, Foxby, Grith, and Reiten 110], and again Foxby 115] the last two articles make use of big Cohen{Macaulay modules. The relationship of injective resolutions to nite free complexes was realized by Peskine and Szpiro in their proof of Bass' conjecture 9.6.2. Their arguments were extended by Foxby 115]. In particular 9.6.1(b) and 9.6.3 (even a more general version for modules) are due to him. As pointed out already, Roberts gave a characteristic free version of 9.6.3. The investigation of 9.6.3 originated from Vasconcelos 380] who proved it for certain one dimensional local rings.
10 Tight closure
The nal chapter extends the characteristic p methods by introducing the tight closure of an ideal, a concept that, via the comparison to a regular subring or overring, conveys the atness of the Frobenius to non-regular rings. It was invented by Hochster and Huneke about ten years ago and is still in rapid development. The principal classes of rings whose de nition is suggested by tight closure theory consist of the F -regular and F -rational rings they are characterized by the condition that all ideals or, in the case of F -rationality, the ideals of the principal class are tightly closed. Under a mild extra hypothesis F -rationality implies the Cohen{Macaulay property. More is true: F -rational rings are the characteristic p counterparts of rings with rational singularities we will at least indicate this connection { a full treatment would require methods of algebraic geometry beyond our scope. Tight closure theory has many powerful applications. Among them we have selected the Briancon{Skoda theorem, whose proof is based on the relationship of tight closure and integral closure, and the theorem of Hochster and Huneke that equicharacteristic direct summands of regular rings are Cohen{Macaulay. 10.1 The tight closure of an ideal Throughout this section we suppose that all rings are Noetherian and of prime characteristic p, unless stated otherwise. Recall from Section 8.2 that I q], q = pe, denotes the q-th Frobenius power of an ideal I , that is, I q] is the ideal generated by the q-th powers of the elements of I equivalently, I q] is the ideal generated by the image of I under the e-fold iteration F e of the Frobenius homomorphism F : R R , F (a) = ap. We reserve the letter q for powers of p for example, we will say `for q 0' when we mean `for q = pe with e 0'. In the following the set R of elements of R that are not contained in a minimal prime ideal of R will play an important r^ole. Note that R is
!
multiplicatively closed. De nition 10.1.1. Let I R be an ideal. The tight closure I of I is the set of all elements x 2 R for which there exists c 2 R with cxq 2 I q] for q 0. One says I is tightly closed if I = I . 378
379
10.1. The tight closure of an ideal
In previous chapters I has denoted the ideal generated by the homogeneous elements in I where I is an ideal in a graded ring. Since there is no danger of confusion, we keep the `traditional' notation for tight closure. The next proposition lists some basic properties of tight closure in particular it behaves as expected for a closure operation. Proposition 10.1.2. Let I and J be ideals in R. Then the following hold: (a) I is an ideal and I J ) I J (b) there exists c 2 R with c(I )q] I q] for q 0 (c) I I = I (d) if I is tightly closed, then so is I : J (e) x 2 I if and only if the residue class of x lies in ((I + )= ) for all
minimal prime ideals of R (f) if R is reduced or height I > 0, then x c R with cxq I q] for all q. p
2
2
p
p
2 I implies that there exists
(a) is obvious. (b) We choose a system y1 . . . ym of generators of I . For each i there exist ci 2 R such that ci yiq 2 I q] for q 0, and therefore c(I )q] I q] for c = c1 cm and q 0. (c) Suppose dxq 2 (I )q] for q 0 with d 2 R . With c as in (b) one then has (cd )xq 2 I q] for q 0. Since cd 2 R , it follows that x 2 I . (d) Note that (I : J )q] I q] : J q]. Thus cxq 2 (I : J )q] for q 0 implies c(xy)q 2 I q] for all y 2 J and q 0. Hence xy 2 I = I for all y 2 J , and therefore x 2 I : J . (e) If x 2 I , then the residue class x belongs to ((I + )= ) since R \ = . Conversely, let 1 . . . n be the minimal prime ideals of R , and suppose x 2 ((I + i )= i ) for all i. Then there exist ci 2 R n i with ci xq 2 I q] + i for q 0. We may assume that ci 2 R : replace ci by ci + c0i where c0i 2 j if and only if ci 2= j . (Such c0i exist since the intersection of some minimal prime ideals is not contained in the union P cidi where of the remaining ones.) In the next step we take d = di 2= i , i Q but di 2 j 6=i j . Now pick r = pf so large that ( 1 m )r] = 0. Then we have Proof.
p
p
p
p
p
p
p
p
p
p
p
p
p
Y p
(di ci )r xrq 2 (I rq] +
r] p
i
)
p
r ] p
j 6=i
j
I rq]
for all i:
This implies d r xq 2 I q] for q 0. Since d 2 R , we conclude x 2 I . (f) If height I > 0, then R \ I 6= , so that c 2 R with cxq 2 I q] for q 0 can be replaced by car with a 2 I and r suciently large. Suppose now that R is reduced. Applying the previous argument for the case of positive height to the residue class rings R= i , where 1 . . . m p
p
p
380
10. Tight closure
are again the minimal prime ideals of R , we nd ci 2 R such that ci xq 2 I q] + i for all q 0. Now we choose d as in the proof of (e) and nd that dxq 2 I q] + 1 m = I q]. Usually the computation of I is very dicult. We give two examples. Examples 10.1.3. (a) Let R1 = kX Y Z ]=(X 2 ; Y 3 ; Z 7 ) where k is a eld of arbitrary characteristic p > 0. Evidently R1 is an integral domain, a complete intersection, and therefore Cohen{Macaulay. Furthermore, the ideal generated by the residue classes of the partial derivatives of X 2 ; Y 3 ; Z 7 is primary to the maximal ideal = (x y z ) (small letters denote residue classes). The Jacobian criterion (for example, see 270], (30.4)) shows that (R1) is a regular local ring for 6= . Especially, R1 is a normal ring by Serre's criterion 2.2.22. Setting deg X = 21, deg Y = 14, and deg Z = 6 makes R1 a positively graded k-algebra with maximal ideal . (All these assertions hold over an arbitrary eld k.) We claim that x 2 (y z ) . If p = 2, then obviously xq 2 (y z )q] for all q = pe. For p > 2 one has cxq 2 (y z )q] for c = x. In fact, set u = (q + 1)=2. Then xq+1 is a k-linear combination of monomials y3v z 7w with v + w = u. It is an elementary exercise that 3v q or 7w q. (b) Let R2 = kX Y Z ]=(X 2 ; Y 3 ; Z 5 ). Then, as in (a), R2 is a normal complete intersection domain. The graduation is now given by deg X = 15, deg Y = 10, and deg Z = 6. We claim that x 2= (y z ) if and only if char k > 7. In this case (y z ) is tightly closed because the only proper ideal of R=(y z ) is generated by the residue class of x. Evidently S = ky z ] is isomorphic to the polynomial ring in two indeterminates over k and R1 is a free S -module with basis 1 x. Therefore every element f 2 R2 has a unique presentation of the form f0 + xf1 with f0 f1 2 ky z ]. The case p = 2 is trivial. So suppose p is an odd prime. As above, set u = (q + 1)=2, choose c 2 R2, and let s and t denote the highest exponents with which y and z respectively appear in c0 and c1 where c = c0 + xc1 with c0 c1 2 ky z ]. One has p
p
p
m
p
m
p
m
xq+1 = c
X u 3v
v +w =u
v
y z 5w :
;
Therefore 2 ( ) only if all the binomial coecients uv for which 3v + s < q and 5w + t < q vanish modulo p. First let p > 7. Then at least one (for p > 30 each) of the following inequalities has an integral solution: (i) 3 p < p < p (ii) 1 p < p < 1 p: 10 3 6 5 e ; 1 If (i) has a solution p , then v = p p and w = u ; v satisfy the inequalities 3v + s < q and 5w + t < q for e 0, q = pe . Since none of the cxq
yq z q
381
10.1. The tight closure of an ideal
;
factors in the `numerator';of uv = u(u ; 1) (u ; v + 1)=v! is divisible by pe , one sees easily that uv is non-zero modulo p. If (ii) has an integral solution, the argument is analogous. This shows cxq 2 (yq z q ) for q 0 is impossible. Second, for p = 7 neither (i) nor (ii) has an integral solution. Nevertheless, there appears exactly one; multiple ; of 7e;1 in the `numerator' as well as in the `denominator' of wu = uv . Therefore it is enough if we can choose w as an integral multiple of 7e;2 in the critical range, and this is possible since 49=6 < 9 < 49=5. The argument showing that x 2 (y z ) for p = 3 and p = 5 is left to the reader. Though R1 and R2 have a very similar structure, there is an invariant distinguishing them: the a-invariant of R1 is non-negative, namely a(R1) = 1, whereas a(R2) = ;1 (see 3.6.14 and 3.6.15 for the computation of ainvariants). Therefore, if k is a eld of characteristic 0, R2 has a rational singularity by the criterion of Flenner 107] and Watanabe 389] whereas the singularity of R1 is non-rational. The connection between rational singularities and tight closure will be discussed in Section 10.3, and we will see that the di erent behaviour of R1 and R2 with respect to tight closure is by no means accidental. Remark 10.1.4. While it is usually not dicult to show that homologically de ned invariants commute with localization or, in the case of a local ring (R ), with -adic completion, tight closure so far has resisted all e orts to establish these properties for it. It is obvious that (I ) (I ) and I R^ (I R^ ) , but the converse inclusions are only known in special cases, some of which will be discussed below. The best result available for localization is due to Aberbach, Hochster, and Huneke 2]: under some mild conditions on R one has (I ) = (I ) for ideals I of nite phantom projective dimension this includes all ideals of nite projective dimension. The de nition of nite phantom projective dimension requires the introduction of tight closure for submodules U M (see Hochster and Huneke 192] and Aberbach 1]). The following proposition indicates how elements in the tight closure of an ideal may arise in a non-trivial way. Proposition 10.1.5. Let S R be a module- nite R-algebra. Then one has (IS ) \ R I for all ideals I of R. Proof. Assume rst that R and S are integral domains. Then there are a free R -submodule F of S and an element e 2 R , e 6= 0, with eS F , and for each element u 2 F , u 6= 0, there exists an R -linear map f : F ! R with f (u) 6= 0. Therefore, given c 2 S , one can nd an R -linear map g : S ! R with g(c) 6= 0. m
m
p
p
p
p
382
10. Tight closure
Now pick x 2 (IS ) \ R . Then there is a c 2 S with cxq 2 (IS )q] = for all q 0, and applying an R -linear map g one gets g(c)xq 2 I q] . Choosing g as above, one concludes x 2 I . In the general case let 1 . . . m be the minimal prime ideals of R , and pick minimal prime ideals 1 . . . m with i = i \ R . (This is possible by A.6.) Let i : R ! R= i be the natural map and 'i the composition R ! R= i ! S= i . Then I q]S
p
p
q
q
p
q
p
p
q
(IS ) \ R
\ i
;
';i 1 (IS )=
i
q
\ ';1(IS= ) ;1(IR= ) i i i i q
p
where the last inclusion is given by the rst part of the proof. By virtue of 10.1.2(e) it follows that (IS ) \ R I . In the next remark and in Section 10.3 we will need the notion of excellence for rings. A Noetherian ring R is called excellent if it satis es the following conditions: (i) R is universally catenary (ii) for all prime ideals of R , all prime ideals of R , and all nite eld extensions L k( ) the ring (R )b L is regular ((R )b is the R -adic completion of R ) (iii) for every nitely generated R -algebra S the singular locus Sing S = f 2 Spec S : S non-regularg is closed in Spec S . Property (ii) is called the geometric regularity of the formal bres of all localizations of R . Complete local rings, and in particular elds are excellent. Furthermore the localizations of an excellent ring R and the nitely generated R -algebras are excellent as well. We refer the reader to 270], x32 or 142], IV.7.8 for a systematic development of this concept. Remarks 10.1.6. (a) Suppose R is a domain and S a module- nite extension domain. Then the eld of fractions of S is an algebraic extension of R and can therefore be embedded into a xed algebraic closure L of the eld of fractions of R . Through this embedding, S is contained in the integral closure R + of R in L one calls R + the absolute integral closure of R . Conversely, R + is the union of module- nite extension domains of R . Thus 10.1.5 implies IR + \ R I . It is not known whether equality holds in general, but Smith 348] has proved that IR + \ R = I for ideals I of the principal class in domains R such that R is excellent for all 2 Spec R. (b) By a remarkable theorem of Hochster and Huneke 193], the ring R + is a big Cohen{Macaulay algebra for R if R is an excellent local domain of characteristic p. This allows one to construct big Cohen{ Macaulay algebras for all Noetherian local rings containing a eld moreover, the construction is `functorial' in the best possible way. See p
q
p
q
p
p
p
p
q
q
p
p
p
383
10.1. The tight closure of an ideal
Hochster and Huneke 197] for the numerous applications of the existence and functoriality of big Cohen{Macaulay algebras. The next theorem gives a crucial property of tight closure. It also shows that the attribute `tight' is well chosen. Theorem 10.1.7. Let R be a regular ring. Then (a) I q] : J q] = (I : J )q] for all ideals I and J of R, and (b) every ideal of R is tightly closed. Proof. (a) By induction it is enough to show I p] : J p] = (I : J )p]. One has I p] = IR F where R F is R viewed as an R -algebra via the Frobenius endomorphism F . For a regular ring R , the R -algebra R F is at by Kunz's theorem 8.2.8, and we show more generally that IS : JS = (I : J )S if S is a at algebra over R . The ideal I : J is the annihilator of the R -module (J + I )=I . Since S is at, one has natural isomorphisms (I : J ) S = (I : J )S and ((J + I )=I ) S = ((J + I ) S )=(I S ) = (J + I )S=IS:
Therefore it is enough to show (AnnR M )S = AnnS (M S ) for a nite R module M . This follows by tensoring the exact sequence 0 ! AnnR M ! R ! EndR (M ) with S and using the natural isomorphism EndR (M ) S = EndS (M S ). (b) Let I be an ideal of R and suppose that cxq 2 I q] for x 2 R , x 2= I , c 2 R , and q 0. Then I : x 6= R , and all the conditions remain true after localization at a prime ideal containing I : x. In order to derive a contradiction we may therefore assume that R is local with maximal ideal . By (a) one has (I : x)q] = I q] : xq for all q 0. Therefore, if c 2 I q] : xq for q 0, then c 2 (I : x)q] q] q for q 0. This implies c = 0, the desired contradiction. For several theorems below it will be essential that R is equidimensional : this means dim R= = dim R < 1 for all minimal prime ideals of R . Corollary 10.1.8. Suppose R is equidimensional and a nite module over a regular domain A. Then IR :R JR ((I :A J )R ) and IR \ JR ((I \ J )R ) m
m
p
for all ideals I and J of A.
m
p
There exist c 2 A, c 6= 0, and a free A-submodule F of R such that cR F . Choose x 2 IR :R JR . Then xq J q] I q] R for all q. Multiplication with c yields J q](cxq ) I q]F . Since F is a free A-module, this implies cxq 2 (J q] : I q] )F . By 10.1.7(a) one has (J q] : I q])F = (I : J )q]F , and so cxq 2 (I : J )q]F (I : J )q]R . The argument for IR \ JR is similar.
Proof.
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10. Tight closure
It remains to show c 2 R for which we need the hypothesis that R is equidimensional. Let be a minimal prime ideal of R . Then dim R= = dim A=( \ A) by the corollary A.8 of the going-up theorem, and there exists such a prime ideal 0 with 0 \ A = 0. Especially, dim A = dim R = dim A=( \A) and, hence, \A = 0 for all minimal prime ideals of R . (Conversely, this fact implies that R is equidimensional.) p
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If, in the situation of 10.1.8, R (and therefore A) is a local ring, then every system of parameters x1 . . . xd of A is also a system of parameters of R and an A-sequence. The last condition is equivalent to (x1 . . . xj ) :A xj +1 = (x1 . . . xj ): Since A is regular, (x1 . . . xj ) = (x1 . . . xj ) for j = 0 . . . d ; 1 by 10.1.7, and so (x1 . . . xj ) :R xj +1 (x1 . . . xj ) j = 0 . . . d ; 1: If R is an equidimensional complete local ring, then we can always nd a suitable regular `Noether normalization' A (see A.22). Roughly speaking one may therefore say that R is `Cohen{Macaulay up to tight closure'. This holds for a larger class of local rings. Theorem 10.1.9 (Hochster{Huneke). Let R be an equidimensional residue class ring of a Cohen{Macaulay local ring A, and x1 . . . xd a system of parameters of R. Then (x1 . . . xj ) :R xj +1
(x1 . . . xj )
j = 0 . . . d
Proof. We write R
; 1:
= A=I . Lemma 10.1.10 below shows that there exists a system of parameters z1 . . . zg y1 . . . yd in A with g = codim I such that z1 . . . zg 2 I and xi is the residue class of yi. Since A is Cohen{Macaulay, z1 . . . zg y1 . . . yd is an A-sequence. Set J = (z1 . . . zg ). Since R is equidimensional, all the minimal prime ideals 1 . . . m of I have height g, and are therefore minimal prime ideals of J . Let m+1 . . . n be the remaining minimal prime ideals of J . If we now choose c 2 ( m+1 \ \ n )s n ( 1 m ) for s suciently large, then cI r J for some r > 0. Furthermore the residue class d of c in R belongs to R . Suppose that bxj +1 2 (x1 . . . xj ) for some b 2 R . Then we pick a preimage a of b in A, obtaining a relation ayj +1 ; (a1 y1 + + aj yj ) 2 I . For q = pe r this entails with buv 2 A: caq yj +1 ; (aq1 y1q + + aqj yjq ) = bq1z1 + + bqg zg However z1 . . . zg y1 . . . yd is an A-sequence, and so is z1 . . . zg y1q . . . ydq (see 1.1.10). Therefore caq 2 (y1 . . . yj )q] + I for all q r, and taking residue classes we get the desired result. p
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10.1. The tight closure of an ideal
Lemma 10.1.10. Let (A ) be a Noetherian local ring (not necessarily of characteristic p), I a proper ideal of A, and x1 . . . xd a system of parameters of A=I. Then one can nd representatives y1 . . . yd of x1 . . . xd in A and z1 . . . zg I, g = codim I, such that z1 . . . zg y1 . . . yd is a system of parameters for A. m
2
Note that g + d = dim A. Suppose we have constructed representatives y1 . . . yd of x1 . . . xd such that codim J = d for J = (y1 . . . yd ). Since dim A=J = g and since (I + J )=J is =J -primary, we can then nd z1 . . . zg 2 I that complement y1 . . . yd to a system of parameters. The elements y1 . . . yd are constructed inductively. Assume that y1 . . . yj ;1 have been found such that codim(y1 . . . yj ;1) = j ; 1. Choose a representative yj0 of xj . Then Proof.
m
dim A=(y1 . . . yj ;1 I yj0 ) = d ; j < g + d ; j + 1 = dim A=(y1 . . . yj ;1):
Thus I + (yj0 ) is not contained in any of the nitely many prime ideals 1 . . . m (y1 . . . yj ;1) with dim A= i = g + d ; j + 1. Now Lemma 1.2.2 (with M = A and N = I + (yj0 )) yields a representative yj of xj such that yj 2= i for i = 1 . . . m. In the case in which R is a residue class ring of a Gorenstein local ring, one can give a shorter proof of 10.1.9, using 8.1.3. This technique will be applied in the proof of 10.4.4. p
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F-regularity. Theorem 10.1.9 shows that rings in which every ideal is
tightly closed have special properties. They deserve a special name. De nition 10.1.11. One says R is weakly F-regular if every ideal of R is tightly closed. If all rings RT of fractions of R are weakly F -regular, then R is F-regular. The distinction between weak F -regularity and F -regularity is undesirable but hard to avoid as long as the localization of tight closure has not been proved. However, it is enough to require F -regularity for the localizations R , 2 Spec R : Proposition 10.1.12. (a) Let I be an ideal primary to a maximal ideal . Then (IR ) = I R . (b) If every ideal primary to a maximal ideal is tightly closed, then R is p
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weakly F-regular. (c) R is weakly F-regular if and only if R is weakly F-regular for all maximal ideals . (d) A weakly F-regular ring is normal. (d) If R is a weakly F-regular residue class ring of a Cohen{Macaulay ring, then R is Cohen{Macaulay. m
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10. Tight closure
(a) We only need to show the inclusion (IR ) I R , and it holds if (IR ) \ R I . By virtue of 10.1.2 it is enough that ((IR ) \ R + )= ((I + )= ) for all minimal prime ideals of R . If 6 , equivalently I 6 , then both sides equal R= . So suppose . Then the image of x 2 (IR ) \ R under the natural map R ! R = R certainly belongs to the tight closure of (I + )R = R . This observation reduces (a) to the case of an integral domain R in which we have R \ R = R . (So far we have only used that R is a localization of R .) Suppose that cxq 2 I q] for x 2 R , c 2 R , and q 0. Then we can obviously assume c 2 R . It follows that c 2 R . Furthermore cxq 2 I q] \ R = I q] where for the last equation we have used that I q] is -primary because Rad I q] = and is a maximal ideal. (b) By Krull's intersection theorem, every ideal I of R is the intersection of the ideals I + n where is a maximal ideal containing I and n 2 N. Furthermore the intersection of tightly closed ideals is tightly closed. (c) is an immediate consequence of (a) and (b). (d) will be proved after 10.2.7. (e) We must show that R is Cohen{Macaulay for all maximal ideals . Part (c) implies that R is weakly F -regular. Thus R is a normal domain by (d) and, therefore, equidimensional. Now the Cohen{Macaulay property results from 10.1.9. The following proposition yields the most important examples of F -regular rings. Proposition 10.1.13. Let S R be a (weakly) F-regular R-algebra such that IS \ R = I for all ideals I of R. If R S , then R is (weakly) Proof.
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F-regular.
The hypothesis '(R ) S implies that (I )S (IS ) , whence the assertion about weak F -regularity is obvious. Furthermore it is inherited by every localization, as is the condition IS \ R = I : if the induced homomorphism R=I ! S=IS is injective, then so is RT =IRT ! ST =IST for all multiplicatively closed subsets T of R , and every ideal of RT has the form IRT for an ideal I of R . The hypothesis IS \ R = I is satis ed if R is a direct summand of S as an R -module or, more generally, if S is pure over R (see 6.5.3(b) for the notion of purity). An immediate corollary is the characteristic p version of the Hochster{Roberts theorem. Corollary 10.1.14. Let the ring R be a direct summand of the regular ring Proof.
S. If R is a residue class ring of a Cohen{Macaulay ring, then R is Cohen{ Macaulay.
387
10.2. The Briancon{Skoda theorem
Exercises
and J be ideals of R . Show (I \ J ) I \ J , (I + J ) = (I + J ) , and (IJ ) = (I J ) furthermore (0) = Rad(0). (b) Let R = R= Rad(0). Prove I is the preimage of (I R ) under the natural homomorphism R ! R . 10.1.16. (a) Let x1 . . . xn y z be elements of R such that the ideals (x1 . . . xn y) and (x1 . . . xn z ) are tightly closed and grade(x1 . . . xn y) = n + 1. Show (x1 . . . xn yz ) is tightly closed. (Use 1.6.17.) (b) Suppose that grade(x1 . . . xn ) = n and (x1 . . . xn ) is tightly closed. Show (xa1 . . . xann ) is tightly closed for all integers a1 . . . an 1. 10.1.17. Find a tight closure proof of the `monomial theorem' 9.2.1 in characteristic p. 10.1.18. (a) Show that a primary component of a tightly closed ideal I that belongs to a minimal prime ideal of I is tightly closed. (Hint: = I : x for a suitable x.) (b) Let I be a tightly closed ideal such that the maximal ideal is a minimal prime ideal of I . Show I is tightly closed. (c) Let I be an ideal all of whose minimal prime ideals are maximal ideals. Show I is tightly closed if and only if all the localizations I with respect to maximal ideals are tightly closed. 10.1.19. (Smith) Prove the following assertions: (a) If tight closure commutes with localization in R= for each minimal prime of of R , then tight closure commutes with localization in R . (b) Let R be a domain that has an F -regular module- nite extension. Then tight closure commutes with localization in R . (c) Tight closure commutes with localization in rings R = k X1 . . . Xn ]=I where I is generated by monomials and binomials. (Hint: the minimal prime ideals of I are again generated by such elements, and if I is prime, then R is an a ne semigroup ring. See Eisenbud and Sturmfels 95] for the theory of binomial ideals.) 10.1.15. (a) Let I
1
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10.2 The Briancon{Skoda theorem This section is devoted to the relationship between the tight closure and the integral closure of an ideal. Our major objective is a proof of the Briancon{Skoda theorem for regular rings containing a eld. It will be derived from its tight closure variant by reduction to characteristic p. Integral dependence on an ideal. We rst discuss the basic notion of integral dependence on an ideal I and introduce the integral closure of I . De nition 10.2.1. Let R be a ring and I R an ideal. Then x R is integrally dependent on I or integral over I if and only if there exists an
equation
xm + a1xm;1 +
+ am = 0
with ai 2 I i i = 1 . . . m:
2
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10. Tight closure
The elements x 2 R that are integral over I form the integral closure I of I . It is evident that I I Rad I , that I1 I2 ) I1 I2, and that integral dependence is preserved under ring homomorphisms. The following proposition lists less obvious properties of integral dependence. Proposition 10.2.2. (a) The following are equivalent: (i) x 2 I (ii) there exists m 1 with xm 2 I (I + Rx)m;1 (iii) there exists m 1 with (I + Rx)m+k = I k+1(I + Rx)m;1 for all k
2 N
(iv) there exists a nite ideal J
R such that xJ IJ and Ann J annihilates a power of x. (b) I is an integrally closed ideal. (c) Suppose that R is Noetherian. Then x I if and only if the residue class of x is integral over (I + )= for all minimal prime ideals of R.
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(a) The equivalence of (i) and (ii) is evident, and (ii) results from (iii) with k = 0. Conversely, xm 2 I (I + Rx)m;1 implies (I + Rx)m = I (I + Rx)m;1 from which (iii) follows by induction on k. For (i) ) (iv) pick x 2 I . Then there exists a nite subideal I 0 of I over which x is integral. Therefore we may assume I to be nite and choose J = Rxm;1 + Ixm;2 + + I m;1 . For (iv) ) (i) let J be generated by y1 . . . yn . Then there exists an n n matrix A = (aij ) with aij 2 I such that (xEn ; A)y = 0 where y is the column vector with components yj and En is the n n unit matrix. It follows that det(xEn ; A)J = 0, and so det(xEn ; A) 2 Ann J . Upon multiplication by a power of x we obtain an equation showing x 2 I . (b) It is obvious that ax 2 I for all x 2 I and a 2 R . Suppose x1 x2 2 I . Again we may assume that I is nitely generated and we choose J1 for x1 and J2 for x2 as we have chosen J for x above especially, both J1 and J2 contain a power of I . It follows immediately that (x1 + x2)J1J2 IJ1 J2 furthermore Ann J1J2 annihilates a power of I and therefore annihilates (x1 + x2)n for n 0. The argument showing that I is integrally closed is similar and can be left to the reader. (c) The `only if' part is obvious. For the `if' part let 1 . . . r be the minimal prime ideals of R . We lift an integral dependence equation of the residue class of x with respect to (I + i )= i to a relation Fi (x) 2 i such that the coecients of the powers of x satisfy the requirements of De nition 10.2.1. Then F (x) = F1(x) Fr (x) 2 1 r and a suitable power of F (x) vanishes. For ideals J I of a Noetherian ring R one has I = J if and only if J is a reduction ideal of I (see Exercise 10.2.10). Proof.
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389
10.2. The Briancon{Skoda theorem
We note a useful criterion for normality. Proposition 10.2.3. A Noetherian ring R is normal if and only if it satis es the following conditions: (i) R is a eld for each prime ideal that is both minimal and maximal (ii) the principal ideals (x), x R , are integrally closed.
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The essential observation relating normality and condition (ii) is the following: let x be a regular element of R and suppose we have an integral dependence relation ym + a1xym;1 + + am;1xm;1 y + amxm = 0 ai 2 R: Then the element y=x of the total ring of fractions Q of R is integral over R . Now, if R is integrally closed in Q, it follows that y=x 2 R and, hence, y 2 (x). Conversely, if f=g (f g 2 R , g a regular element) is integral over R , one sees immediately that f is integral over the ideal (g). Suppose now that R is normal. Then it is the direct product of nitely many integrally closed domains. Therefore it obviously satis es condition (i). Furthermore every element x 2 R is a regular element of R so that the previous observation immediately yields that (x) is integrally closed. For the converse we rst split R into a direct product R1 Rr such that Spec Ri is irreducible for each of the rings Ri . It suces to show that each Ri is a normal domain. Note that condition (ii) is inherited by Ri. Furthermore condition (i) implies that Ri is a eld if Ri has a prime ideal that is both minimal and maximal. So we can assume that Spec R is irreducible and R has no such prime ideal. The rst (and crucial) step is to show that R is reduced. For each 1 . . . s, there is a non-minimal prime ideal minimal prime ideal Ti , i = S in i i . Choose a 2 ( i i ) n ( i i ). The nilradical N of R is contained T every integrally closed ideal, and therefore it is contained in j (aj ) since aT2 R . There exists an element c 2 R such that b = 1 ; ca annihilates j j (a ) (this is the usual argument from which Krull's intersection theorem is derived). A fortiori, bN = 0. The choice of a ensures that b 2 R as well, and, by the same token, we have (1 ; db)N = 0 for some d 2 R . This shows N = 0. Since R is reduced, the total ring of fractions Q of R is the direct product of elds Qi . The idempotents ei representing the unit elements of Qi satisfy the equation e2i ; ei = 0. Write ei = fi =gi with fi 2 R and a regular element gi 2 R . The initial observation yields ei 2 R . By the assumption on R this is only possible if Q is a eld and, hence, R is a domain. Now we apply the initial observation once more to conclude that R is integrally closed. For the connection with tight closure it is important that in a Noetherian ring integral dependence can be characterized by homomorphisms Proof.
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390
10. Tight closure
to valuation rings. Let K be a eld. Recall that a proper subring V of K is a valuation ring of K if x 2 V or x;1 2 V for all x 2 K . It follows that the set of ideals of V is linearly ordered by inclusion in particular V is local and every nite ideal of V is principal. If V is Noetherian, then the maximal ideal V of V is principal, and conversely a Noetherian valuation ring V is a regular local ring of dimension 1 and is termed a discrete valuation ring. The following theorem will be crucial: let K be a eld, A a subring of K , and A a prime ideal of A then there exists a valuation ring V of K such that A V and V \ A = . Furthermore it is easily proved that a valuation ring is normal. (See 270], x10, 47], Ch. VI, or 397], Vol. II, Ch. VI for proofs and more information on valuation rings.) Proposition 10.2.4. (a) Let R be an integral domain with eld of fractions K and I an ideal of R. Then I is the intersection of all ideals IV where V m
p
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ranges over the valuation rings of K containing R. (b) Suppose R is a Noetherian ring. Then there exist a nite number of homomorphisms 'i from R to discrete valuation rings Vi such that Ker 'i is a minimal prime ideal of R and I is the intersection of the preimages ';1(IVi ).
(a) Let J be the intersection of the ideals IV . For I J it is enough that all the ideals IV are integrally closed. As observed above, IV is a principal ideal of V . Since V is a normal domain, principal ideals of V are integrally closed (see 10.2.3 { for this implication the Noetherian property is irrelevant). For the converse inclusion choose x 2 J . Let L be the set of all quotients a=x with a 2 I and consider the ideal LR L] in the subring R L] of K . If LR L] were a proper ideal of R L], then there would exist a valuation ring V of K with LR L] V . In particular we would have a=x 2 V for all a 2 I this implies x=a 2= V , and so x 2= IV , a contradiction. Thus LR L] = R L], and there exists a representation 1 = f (a1=x . . . am =x) where f is a polynomial with coecients in R and a1 . . . am 2 I . Multiplication by a suciently high power of x yields an integral dependence relation for x on I . (b) In view of 10.2.2 we can restrict ourselves to the case of a domain R . Choose a system of generators x1 . . . xn of I and let Ri beTthe integral closure of R xj =xi : j = 1 . . . n] in K . We claim that I = i Ri xi . The inclusion `' holds because the principal ideal Ri xi is integrally closed in the normal domain Ri . For the converse we use part (a). Let V be a valuation ring of K containing R , and pick an index i such that xi generates IV . Then xj =xi 2TV for all j and therefore Ri xi IV . It follows that the intersection i Rixi is contained in every ideal IV where V ranges over the valuation rings of K . Though the ring Ri need not be Noetherian, it is a Krull ring (see Proof.
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10.2. The Briancon{Skoda theorem
270], x12). A divisorial ideal of a Krull ring, T and especially a principal ideal, has the primary decomposition = j ( R j \ R ) where the j are the nitely many divisorial prime ideals containing , and furthermore R j is a discrete valuation ring. a
a
a
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Tight closure and integral closure. After these preparations we can easily
show that tight closure is tighter than integral closure. In the sequel we shall again assume that R is a Noetherian ring of characteristic p. Proposition 10.2.5. One has I I for all ideals I of R. Proof. Let ' be a homomorphism from R to a discrete valuation ring V such that Ker ' is a minimal prime ideal of R . Then '(I )V (IV ) since '(R ) V . Moreover, V is a regular local ring and, thus, (IV ) = IV . So 10.2.4 implies I I . It is easy to give examples of tightly closed ideals that are not integrally closed. For example, every ideal in a polynomial ring R over a eld is tightly closed, but not every ideal of R is integrally closed if dim R > 1 (see Exercise 10.2.12). The tight closure version of the Briancon{Skoda theorem is an `asymptotic' converse of the previous proposition. Theorem 10.2.6 (Hochster{Huneke). Let I be an ideal of R generated by elements f1 . . . fn. (a)Then I n+w (I w+1) for all w 2 N. (b) If R is regular or just weakly F-regular, then I n+w I w+1, and in particular I n
I.
We must relate Frobenius powers and ordinary powers of I . This is possible through the equation I k(n+w) = (f1k . . . fnk )w+1I k(n;1) whose elementary veri cation is left to the reader. In view of 10.1.2 and 10.2.2 we may assume that R is an integral domain. Set J = I n+w and pick x 2 J . By 10.2.2 there exists m 1 with (J + Rx)m+k = J k+1(J + Rx)m;1 for all k 2 N in particular xm xk 2 J k = I k(n+w) (f1k . . . fnk )w+1I k(n;1) for all k 2 N. Setting c = xm and k = q = pe we obtain cxq 2 (f1q . . . fnq )w+1I q(n;1) (I w+1)q] as desired. Part (b) results immediately from (a). The following corollary is crucial for issues of normality. Proof.
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10. Tight closure
Corollary 10.2.7. Let I = (x) be a principal ideal. Then I = I . Proof. The inclusion I I is Proposition 10.2.5, and the converse inclusion is contained in the theorem. As a consequence of 10.2.3 and 10.2.5 we derive the normality of a weakly F -regular ring R , which has already been stated in 10.1.12. Since Rad(0) = (0), an F -regular ring is reduced, whence it satis es condition (i) of 10.2.3. By de nition it also ful lls condition (ii) so that normality follows immediately. The original Briancon{Skoda theorem 347] is essentially the assertion of 10.2.6 for R = ChX1 . . . Xd i, the ring of convergent power series in d indeterminates. It was motivated by the following problem: given f 2 (X1 . . . Xd ), what is the smallest number m such that f m 2 I = (X1@1f . . . Xd @d f )? (Here @i is the partial derivative with respect to Xi.) Answering this question obviously generalizes the well-known rule uf = X1@1 f + + Xd @d f for a homogeneous polynomial f of degree u. The connection with integral closure is given by the fact that f 2 I this results easily from the criterion 10.2.4. We derive a generalization of the original Briancon{Skoda theorem from 10.2.6 by reduction to characteristic p. Theorem 10.2.8 (Lipman{Sathaye). Let R be a regular ring containing a eld of arbitrary characteristic and I be an ideal of R generated by elements f1 . . . fn. Then I n+w I w+1 for all w N, and in particular I n I.
2
Proof. The theorem has already been proved in characteristic p. So suppose that there exists a counterexample (R f1 . . . fn) in characteristic 0. Suppose that y 2 I n+w but, y 2= I w+1. Then there is a maximal ideal of R such that x 2= I w+1, and since integral closure commutes with localization (see Exercise 10.2.10), we may assume R is local. For the application of the `regular' variant (b) of Theorem 8.4.1 we must show that our data have a regular equational presentation. That y 2 I n+w can easily be expressed in terms of a single equation: we simply choose indeterminates representing y, the generators of I , and the coecients in an integral dependence relation for y on I n+w. The dicult part of the problem, namely to express the condition y 2= I w+1, has fortunately been solved in Corollary 8.4.4. (Observe that the generators of I w+1 are polynomials in f1 . . . fn .) It follows that there exists a counterexample to the theorem in which R is a regular local ring of characteristic p > 0, a contradiction to 10.2.6. m
m
Remarks 10.2.9. (a) If (R ) is a local ring with an in nite residue class eld, then every ideal I has a reduction ideal J I generated by at most d = dim R elements see 4.6.8. Since J w is a reduction of I w for m
m
393
10.3. F-rational rings
all w 2 N, one can replace n by the minimum of n and d in 10.2.6 and 10.2.8 if the hypothesis on R is satis ed. (b) Theorem 10.2.8 was proved by Lipman and Sathaye 260] for arbitrary regular rings. A variant, valid for an ideal I generated by a regular sequence in a pseudo-rational ring (see 10.3.23 below), was given by Lipman and Teissier 261]. (c) Since it seems impossible to derive the mixed characteristic cases of these theorems from characteristic p results, the tight closure approach does not supersede the proofs given by Lipman{Sathaye and Lipman{ Teissier. However it o ers a re nement we have neglected so far, namely the extra factor I q(n;1) that appears in the proof of 10.2.6. Taking care of it leads one to the Briancon{Skoda theorems with coecients of Aberbach and Huneke 4]. Exercises
(a) Let I and J be ideals in a ring R and x 2 R integral over I , y 2 R integral over J . Deduce xy is integral over IJ . (b) Show that x is integral over the ideal I if and only if xt 2 R t] is integral over the Rees algebra R(I ) = R It]. (Thus integral dependence on ideals can be considered a special case of integral dependence on rings.) (c) Let J I be ideals and suppose I is nitely generated. Prove that I = J if and only if there exists r 2 N with JI r = I r+1. (d) Let T R be a multiplicatively closed set and S = T 1 R . Show IS = I S . 10.2.11. The de nition of integral dependence can be extended as follows: let R S be rings and I R an ideal then x 2 S is integral over I if it satis es an equation as in 10.2.1. Extend 10.2.2 and 10.2.10 to this situation. 10.2.12. Let K be an arbitrary eld and I K X1 . . . Xn ] an ideal generated by monomials. Show that the integral closure of I is the ideal generated by all monomials whose exponent vector belongs to the convex hull (in Rn or Qn ) of the set of exponent vectors of the monomials in I . 10.2.13. Given a regular local ring of dimension n, nd an n-generated ideal I of R with I n 1 6 I . 10.2.10.
;
;
10.3 F -rational rings Throughout this section we suppose that all rings are Noetherian and of characteristic p, unless stated otherwise. Recall that in a weakly F -regular
ring every ideal is tightly closed by de nition. Now we discuss a weaker condition for a ring R :
De nition 10.3.1. One says R is F-rational if the ideals of the principal class, that is, ideals I generated by height I elements, are tightly closed.
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Note that for an equidimensional, universally catenary local ring (R ) an ideal I = (x1 . . . xi ) is of the principal class if and only if x1 . . . xi are part of a system of parameters of R . The name `F -rational' indicates that such rings are the characteristic p analogues of rings with rational singularities. The results of Smith 349] and Hara 149] discussed at the end of this section justify this comparison. In the following we present some basic properties of F -rational rings. Just as for weakly F -regular rings it results from 10.2.3 and 10.2.7 that m
Proposition 10.3.2. F-rational rings are normal. The following lemma is essential in the study of F -rational local rings. Proposition 10.3.3. Let (R ) be an equidimensional local ring that is a homomorphic image of a Cohen{Macaulay ring, and (x1 . . . xd ) a system of parameters of R. Then (a) (x1 . . . xi;1) : xi = (x1 . . . xi;1) for all i = 1 . . . d (b) If (x1 . . . xd ) is tightly closed, then so is (x1 . . . xi ) for all i = 1 . . . d. m
Set Ji = (x1 . . . xi) and pick r 2 Ji;1 : xi . Then rxi 2 Ji;1, and hence there exists c 2 R such that c(rxi )q 2 Ji;q]1 for q large see 10.1.2(b). Thus from 10.1.9 we conclude that crq 2 Ji;q]1 : xqi Ji;1, which yields r 2 Ji;1. This proves (a). We derive (b) by descending induction on i. Suppose it is already known that Ji is tightly closed. Let r 2 Ji;1 then r 2 Ji , and hence r 2 Ji by the induction hypothesis. So r = a + xi b with a 2 Ji;1 and b 2 R . Then r ; a 2 Ji;1, whence b 2 Ji;1 : xi = Ji;1 by (a). This shows Ji;1 = Ji;1 + xi Ji;1, and the conclusion follows from Nakayama's lemma. Proof.
Corollary 10.3.4. An F-rational ring R is Cohen{Macaulay if it is a homomorphic image of a Cohen{Macaulay ring.
2
Proof. Let be a maximal ideal of R . We choose elements x1 . . . xd , d = dim R that generate an ideal I of the principal class. Especially, x1 . . . xd form a system of parameters in R . By hypothesis I is tightly closed. As is a minimal prime ideal of I , we conclude from 10.1.18 that I is tightly closed. Notice that R is a domain since it is normal by 10.3.2. Hence 10.3.3(b) entails that the ideals (x1 . . . xi )R are tightly closed for all i. Now 10.3.3(a) and 10.1.9 imply that x1 . . . xd is an R -sequence. Thus R is Cohen{Macaulay. m
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For local rings, F -rationality is easier to control. In fact one has
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Proposition 10.3.5. Let (R ) be a local ring that is a homomorphic imm
age of a Cohen{Macaulay ring. Then R is F-rational if and only if it is equidimensional and one ideal generated by a system of parameters is tightly closed.
Let x1 . . . xd be a system of parameters of R generating a tightly closed ideal. By 10.3.3(b), R is F -rational if any other system of parameters y1 . . . yd of R generates a tightly closed ideal P as well. Choose t 2 N such that y1 . . . yd 2 (xt1 . . . xtd ), and write yi = dj=1 aij xtj . Then (y1 . . . yn) = (xt1 . . . xtd ) : a with a = det(aij ). This follows from 2.3.10, since, by 10.3.4, R is Cohen{Macaulay, so that every system of parameters is R -regular. Now Exercise 10.1.16(b) tells us that (xt1 . . . xtd ) is tightly closed. Finally 10.1.2 implies that (y1 . . . yd ) is tightly closed, too. Proposition 10.3.6. Let R be a homomorphic image of a Cohen{Macaulay Proof.
ring. Then R is F-rational if and only if R is F-rational for every maximal ideal of R. m
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`(': Let I R be an ideal of the principal class. Suppose that I is strictly contained in I . Then for some maximal ideal of R we have again a strict inclusion I (I ) . It follows that I is not tightly closed as (I ) (I ). This is a contradiction since I is of the principal class, and R is F -rational. `)': Let be a maximal ideal of R . As in the proof of 10.3.4 we conclude that some ideal in R generated by a system of parameters is tightly closed. Hence the assertion results from 10.3.5. Now we can easily show that for a Gorenstein ring `F -rational' is a condition as strong as `F -regular'. Proposition 10.3.7. A Gorenstein ring is F-regular if and only if it is FProof.
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rational. Proof.
In view of 10.3.6 and 10.1.12(c) we only need to show that an
F -rational Gorenstein local ring is F -regular. In order to apply 10.1.12(b), we choose an ideal I which is primary to the maximal ideal of R and show it is tightly closed. There exists an ideal J I generated by a system of parameters. Since R is Gorenstein, we have I = J : (J : I ).
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(This follows immediately from Exercise 3.2.15 applied to the Artinian ring R=J ). The ideal J is tightly closed since R is F -rational, and by 10.1.2(d), I is tightly closed as well. The previous proposition cannot be generalized essentially there exist F -rational, but not weakly F -regular rings of dimension 2 see Watanabe 390] or Hochster and Huneke 196], (7.15), (7.16). F -rationality has good permanence properties for example, it localizes as will easily follow from
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Proposition 10.3.8. Suppose I is an ideal generated by an R-sequence. Then (IRS ) = I RS for every multiplicatively closed set S R. The proof of the proposition uses Lemma 10.3.9. Let R be an arbitrary Noetherian ring, I R an ideal, and
R a multiplicatively closed set. all m 2 N.
S
(a) Then there exists an element s 2 S such that
S
w2S
I m : w = I m : sm for
(b) Suppose in addition that char R = p > 0, and that I is generated by an R-sequence x1 . . . xn. Then, with s 2 S as in part (a), we have q] I : w = I q] : s(n+1)q for all q: w 2S
(a) It is enough to show the inclusion `'. Let T be the associated graded ring grI (R ). Since T is Noetherian, there exists s 2 S such that AnnT (s) = AnnT (ws) for all w 2 S . (One chooses an element s for which Ann(s) is maximal.) Now suppose u 2 I m : w, u 2 I r n I r+1. We may assume that r < m, since otherwise the assertion is trivial. We claim that usm;r 2 I m . Indeed, uw 2 I m I r+1, and so uws 2 I r+1. By the choice of s this implies us 2 I r+1. Induction on r concludes the proof of (a). (b) Again only the inclusion `' needs proof. Given w 2 S and q, x u 2 I q] : w. We shall prove by induction on h 2 N that the element dh = sq+hu belongs to I q] + I q+h. Once we know this, it follows for h = qn that s(n+1)qu 2 I q] + I q(n+1) = I q] . (The last equality holds, since I q(n+1) I q].) We start the induction with h = 0. Then I q] + I q = I q , and the assertion follows from (a). Now suppose that dh 2 I q] + I q+h for some h > 0. Say, X X dh = ri0 xqi + ra xa a = (a1 . . . an) 2 Nn
Proof.
a
i P with i ai = q + h and ai < q for every i. As wu 2 I q], we get an equation X 00 q X 0 q X a i
ri xi =
i
wri xi +
a
wra x
with certain ri00 2 R . This implies X 00 0 q X a (wri ; ri )xi + wra x = 0: i
a
Since x1 . . . xn is R -regular, all the wra are in I .PThereforePsra 2 I for all a, and we conclude that dh+1 = sdh = dh = i sri0 xqi + a sra xa lies in I q] + I q+(h+1). Indeed, the rst sum belongs to I q], the second to I q+(h+1).
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10.3.8. Let u=1 2 (IRS ) then there exists an element c 2 R such that for all q 0 one nds s(q) 2 S with s(q)cuq 2 I q] . It follows that cuq 2 I q] : s(q). With s 2 S as in 10.3.9(b) we have s(n+1)qcuq = c(sn+1u)q 2 I q]. This implies sn+1u 2 I , and so u=1 2 I RS . The other inclusion is trivial. Now we can show Proposition 10.3.10. Let R be an F-rational ring that is a homomorphic Proof of
image of a Cohen{Macaulay ring, and S a multiplicatively closed set in R. Then RS is F-rational. Proof. By 10.3.6 it suces to show that R is F -rational for every prime ideal of R . Since R is Cohen{Macaulay, we have height = grade( R ) (see 2.1.4). Therefore there exists an R -sequence x1 . . . xd of length d = height . In R this sequence forms a system of parameters. By 10.3.5 we only need that (x1 . . . xd )R is tightly closed. But this results from p
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10.3.8. Another easily proved permanence property is given by the following Proposition 10.3.11. Let (R ) be a local ring, and let x 2 be an Rp
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regular element. Then R is F-rational, if R=xR is F-rational. Proof. R=xR is Cohen{Macaulay by 10.3.4, and so is R . In particular R is equidimensional. We may extend x to a system of parameters x x2 . . . xd of R . According to 10.3.5 it suces to show that I = (x x2 . . . xd ) is tightly closed. Choose u I and c R such that cuq I q] for q 0. We may write c = dxt where d xtR for some t. Then duq (xq;t xq2 . . . xqd ) for q 0. Since d = 0 and since the F -rational ring R=xR is a domain, the image of u in R=xR is in the tight closure of (x2 . . . xd )R=xR . Since this ideal is tightly closed, u I as desired.
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At this point it is useful to resume the discussion of the examples 10.1.3. Examples 10.3.12. (a) We have seen that x 2 (y z ) for R1 = kX Y Z ]= (X 2 ; Y 3 ; Z 7 ) where k is a eld of positive characteristic. Therefore R1 is not F -rational, independently of k. Moreover, no ideal I generated by a system of homogeneous parameters of R1 is tightly closed. Otherwise I would be tightly closed in the localization (R1) with respect to = (x y z ), and it would follow that (R1) is F -rational. Since (R1) is regular for prime ideals 6= , the ring R1 would have to be F -rational, too. That R1 is not F -rational follows also from the fact that a(R1) 0 see Exercise 10.3.28. (b) We have also seen that (y z ) is tightly closed in R2 = kX Y Z ]= (X 2 ; Y 3 ; Z 5 ) where k is a eld of characteristic at least 7. Therefore (R2) is F -rational, and so is R2, by the same localization argument as in (a). Since R2 is Gorenstein, it is even F -regular. m
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10. Tight closure
Remarks 10.3.13. (a) Let R be a positively graded ring with maximal ideal . In analogy to the assertions relating the homological properties of R and R (for example, see 2.1.27), one may ask whether the (weak) F -regularity or F -rationality of R and that of R are equivalent. At least for F -rationality there is a satisfactory theorem: a homogeneous k-algebra R over a perfect eld is F -rational if and only if R is F -rational. For weak F -regularity there is only a weaker result. See Hochster and Huneke 196], (4.7) and (4.6). (b) Again in analogy to the homologically de ned ring-theoretic properties, one may ask how (weak) F -regularity and F -rationality behave under at ring extensions with `good' bres. We refer the reader to Hochster and Huneke 195] and Velez 385] for theorems of this type unfortunately they are much harder to prove than their homological counterparts. m
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Test elements. The proofs of the next results require test elements. We briey discuss this notion. De nition 10.3.14. An element c 2 R is called a test element if for all ideals I and all x 2 I one has cxq 2 I q] for all q. The following is the most general existence theorem for test elements. Theorem 10.3.15 (Hochster{Huneke). Let R be a reduced algebra of nite type over an excellent local ring (S ). Let c 2 R be an element such that n
Rc is F-regular and Gorenstein. Then some power of c is a test element.
The theorem implies in particular that test elements exist in reduced excellent local rings: choose an element c 2 I \ R where I is an ideal with Sing R = V (I ). For the proof of 10.3.15 the reader is referred to 195], (6.2). We will show the existence of test elements only in the important special case of reduced F - nite rings: one calls R F- nite if R , viewed as an R -module via F , is nite. For example, every ring which is a localization of an ane algebra over a perfect eld and every complete local ring with perfect residue class eld is F - nite. By a theorem of Kunz 248], F - nite rings are excellent. Theorem 10.3.16. Let R be an F- nite reduced ring, and c 6= 0 an element
of R such that Rc is regular. Then some power of c is a test element. Let R be a domain of characteristic p with quotient eld K for each integer e one may then identify R , viewed as an R -module via F e , with the ring R 1=q, q = pe , of the q-th roots of the elements of R in some algebraic closure of K . The R -algebra structure of R 1=q is of course given by the inclusion map R R 1=q. The notation R 1=q is convenient in the
next lemma that will be needed for the proof of 10.3.16.
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10.3. F-rational rings
Lemma 10.3.17. Let R be an F- nite regular domain, and d 2 R. Then there exist a power q of p and an R-linear map ' : R 1=q ! R such that '(d 1=q) = 1. Proof. Krull's intersection theorem implies that for each maximal ideal there is a power q of p with d 62 q ] . By Kunz's theorem 8.2.8, 1=q R is a free R -module. Since d 1=q =1 is part of a basis of R 1=q , there exists an R -homomorphism : R 1=q ! R with (d 1=q =1) = 1. The map is of the form =a where : R 1=q ! R is R -linear and a 2 R n in particular (d 1=q ) = a . Since the ideal generated by the elements a is not contained in a maximal ideal, it is the unit ideal, and hence 1 is aPlinear combination of some elements a1 = a . . . ar = a r , say 1 = bi ai . Set i = i , qi = q i , and q = maxfq1 . . . qr g. Since d 2= i qi] , a fortiori d 2= i q]. Running through the argument above once more, we P may in fact assume that qi = q for all i and i (d 1=q) = ai . Now ' = bi i has the desired property. Proof of 10.3.16. As Rc is regular, the previous lemma implies that there exist a power q of p and an Rc -linear map : Rc1=q ! Rc with (1) = 1. One can write = =cn where : R 1=q ! R is R -linear. It follows that (1) = cn for some n, and replacing c by cn we may as well assume that (1) = c. Restricting to R 1=p (which is contained in R 1=q) yields an R -linear map ' : R 1=p ! R with '(1) = c. We claim that c2 is a test element if char R 6= 2, and that c3 is a test element if char R = 2. In fact, let I R be an ideal of R , and pick x 2 I . Then there exists an element d 2 R with dxq 2 I q] for all q. As before, we nd a power q0 of p and an R -linear map : R 1=q0 ! R such0 that (0 d 1=q0 ) = cN for some N . Taking the q0 -th root of the relation 0 q qq qq ] 1 =q dx 2 I , one obtains d x 2 I q] for all q. Now we apply and get cN xq 2 I q] for all q. Let N be the smallest integer with this property and write N = mp + r with 0 r < p. Then (cr )1=pcm xq 2 I q]R 1=p, and multiplication by (cp;r )1=p yields cm+1xq 2 I q]R 1=p for all q. Applying the linear map ' constructed in the rst paragraph of the proof we obtain cm+2xq 2 I q] for all q. Since N was chosen minimal, m + 2 N . This implies that N 2, if p > 2, and N 3, if p = 2. Using test elements we can now prove a result about the behaviour of tight closure under completion. Proposition 10.3.18. Let (R ) be an excellent local ring with -adic com^ and let I be an -primary ideal of R. Then I R^ = (I R^ ). pletion R, Proof. We denote by Rred the residue class ring of R modulo its nilradical N = Rad(0). Choose an element c 2 R such that (Rred )c is regular (this m
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10. Tight closure
is possible since Sing Rred is Zariski-closed). It follows from 10.3.15 that some power of c is a test element. Replacing c by this power we may assume that c itself is a test element. Let q0 be such that N q0 = 0. Then for all ideals J in R , c has the property that x 2 J if and only if cxq 2 J q] for all q q0 . One therefore says that c is a q0 -weak test element for R . Since R is excellent, the ring (R^ red )c is also regular (this uses the regularity of the formal bres of Rred ), and hence we may assume c is a q0 -weak test element for R^ as well. The inclusion I R^ (I R^ ) is obvious since R (R^ ). For the proof of the other inclusion we rst notice that (I R^ ) (I R^ ), so that it suces to show that (I R^ ) I R^ . We may therefore assume that I is tightly closed. Since (I R^ ) is ^ -primary, there is an ideal J R containing I with J R^ = (I R^ ) . Suppose I R^ is not tightly closed. Then the inclusion I J is proper. Hence there exists an element x 2 ((I R^ ) n I R^ ) \ R . For all q q0 we then have cxq 2 I q] R^ \ R = I q] . This implies x 2 I = I , a contradiction. Corollary 10.3.19. Let (R ) be an excellent local ring. Then R is Frational if and only if its -adic completion R^ is F-rational. Proof. Suppose R is F -rational, and let I be an ideal of R^ generated by a system of parameters. Then there exists an ideal J of R with I = J R^ such that J is also generated by a system of parameters. By 10.3.18 and our assumption, I = (J R^ ) = J R^ = J R^ = I . Conversely, assume that R^ is F -rational and I is an ideal of R generated by a system of parameters. Since R^ is a faithfully at R module, I = (I R^ ) \ R (I R^ ) \ R = (I R^ ) \ R = I . m
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The Frobenius and local cohomology. We shall see that the Frobenius homomorphism F : R R induces a natural action on local cohomology. This leads to an important characterization of F -rationality in terms of
!
local cohomology discovered by Smith. Let x1 . . . xd be a system of parameters of R . We know from Section 3.5 that local cohomology may be computed as the homology of the % complex modi ed Cech C : 0 ;! C 0 ;! C 1 ;! ;! C n ;! 0 M Ct = Rxi xi xit : .
1
1 i1 0 see 152], Theorem III.3.7. The homomorphism is the composition of the maps m
m
n OX ) ;! H d;1(W n E OW ) ;! HEd (W OW ) where is the edge homomorphism E2d ;10 ! E d ;1 of the Leray spectral H d (R ) = H d ;1 (X x m
sequence (see Godement 128], II.4.17)
n
E2pq = H p (X x R q
OX ) ) Epq = H p+q(W n E OW )
and is a connecting homomorphism in the long exact sequence (1). Since is de ned naturally, it has good functorial properties. Suppose R is of characteristic p then we have a morphism of schemes F : W ! W , the absolute Frobenius morphism. This map is the identity on the underlying topological space and the p-th power map locally on sections of OW ! F OW = OW . This morphism of schemes induces a natural map on cohomology with supports, compatible with . In other
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10. Tight closure
words, one has a commutative diagram H d (R ) ;;;;!
?? y
H d (R )
m
HEd (W
m
?? y
OW ) ;;;;! HEd (W OW )
where the top horizontal map is just the Frobenius action on H d (R ) de ned above. Consequently, the kernel of is an F -stable submodule of H d (R ). This observation is part of the proof of Corollary 10.3.24 (Smith). Let (R ) be an excellent local ring of characm
m
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teristic p. If R is F-rational, then it is pseudo-rational.
Proof.
H d (R ).
By 10.3.22 it suces to show that the kernel of is not all of
We may assume that d 2, and prove that codim Ker 2 (which of course implies Ker 6= H d (R )). From the exact sequence (1) we obtain that Ker is the image of : H d ;1 (W OW ) ! H d ;1 (W n E OW ). Since R is normal and is birational, is an isomorphism at primes of height 1 hence H q (W OW ) = 0 for q > 0 and all 2 X with height 1. This implies codim Ker 2. It remains to show that codim Ker 2. Pick 2 X with height = s then (R q OW ) = H q (W OW ) = H q ( ;1(Spec R ) OW ) = 0 for q > 0, q s. In fact, by Chow's Lemma (152], Exercise III.4.10) we can assume that : ;1 (Spec R ) ! Spec R is projective, and is therefore obtained by blowing up an ideal of R . So the maximal dimension of the closed bre of is bounded by s ; 1, whence the assertion on the vanishing of (R q OW ) follows from the comparison theorem for projective morphisms (152], Corollary III.11.2). The vanishing of (R q OW ) for q s implies dim Supp(R q OW ) d ; q ; 1. Hence dim Supp(R q OW ) \ (X n x) d ; q ; 2 and so H p (X n x R q OW ) = 0 for p + q > d ; 2, q > 0. The Leray spectral sequence now yields H d ;1 (W n E OW ) = E1d ;10 . In particular, may be identi ed with the map E2d ;10 ! E1d ;10 which is the composition of the surjective maps m
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;! E3d;10 ;! E4d;10 ;! ;10 is the image of where, for each r 2, the kernel of dr : Erd ;10 ! Erd+1 Erd ;1;rr;1 ! Erd ;10. Since each Erd ;1;rr;1 is a subquotient of E2d ;1;rr;1 = H d ;1;r (X n x R r;1 OW ) E2d ;10
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10.3. F-rational rings
and since codim Supp(R r;1 OW ) 2, as observed above, we conclude codim(Ker dr ) 2 for all r 2. Therefore codim Ker 2. The following corollary is the characteristic p analogue of Boutot's theorem 50] that a direct summand of a rational singularity is a rational singularity. Corollary 10.3.25. Let (R ) be an excellent local ring of characteristic p m
which is a direct summand of an F-regular overring. Then R is pseudorational. Proof. We know from 10.1.13 that a direct summand of an F -regular ring is again F -regular, and hence F -rational. Now we apply 10.3.24.
For the sake of completeness we quote without proofs the extension of the theory to characteristic 0. De nition 10.3.26. Let k is a eld of characteristic 0, and R an ane kalgebra. The ring R is of F-rational type if there exists a nitely generated Z-subalgebra A of k and a nitely generated A-algebra RA, with at structure map A ! RA such that (a) (A ! RA) A k is isomorphic to k ! R (b) the ring RA A A= is F -rational for all maximal ideals in a dense open subset of Spec A. A typical situation described in the de nition is the following: R is an ane k-algebra kX1 . . . Xn]=(f1 . . . fm ) where the polynomials fi are de ned over Z, ZX1 . . . Xn]=(f1 . . . fm ) is a free Z-module, and (Z=p)X1 . . . Xn]=(f 1 . . . f m ) is F -rational for all but nitely many prime elements p. Let X be a scheme of nite type over a eld of characteristic zero. One says that a point x 2 X has F-rational type if x has an open ane neighbourhood de ned by a ring R of F -rational type. The scheme X has F-rational type if every point x 2 X has F -rational type. The following fundamental theorem relates rational singularities and F -rational rings: Theorem 10.3.27 (Smith and Hara). Let X be a scheme of nite type over a eld of characteristic 0. Then X has F-rational type if and only if X has m
m
rational singularities.
Exercises 10.3.28. Let R be a positively graded k-algebra where k is a eld of positive charac-
teristic. Prove that a(R ) < 0 if R is F -rational. (Hint: a(R ) = maxfi : H (R ) 6= 0g. See 196], (7.12) and (7.13) for converse results.) 10.3.29. Show that F -rationality implies F -injectivity for Cohen{Macaulay local rings.
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10. Tight closure
One says that R is F-pure if R is a pure extension of R via the Frobenius map F (see 6.5.3(b) for this notion). Show that k X1 . . . Xn ]=I is F -pure for every eld k of positive characteristic and each ideal I generated by squarefree monomials indeed, R is a direct summand of R under F . 10.3.31. (a) Let R be an arbitrary ring and S a pure extension of R . Show that for every complex C. of R -modules the natural map Hi (C. ) ! Hi (C. S ) is injective for all i. (b) Prove that F -purity implies F -injectivity. (One can show that weak F -regularity implies F -purity see Fedder and Watanabe 104], 1.6.) 10.3.30.
10.4 Direct summands of regular rings In this section we return to a subject that has been treated several times before, namely the Cohen{Macaulay property of direct summands of regular rings, which we will now prove for rings containing a eld { the general case seems to be unknown. The next theorem generalizes 6.5.2, in which we have considered graded direct summands of polynomial rings kX1 . . . Xn], and 10.1.14, which covers rings of characteristic p. Theorem 10.4.1 (Hochster{Huneke). Let R be a Noetherian ring containing a eld and suppose R is a direct summand of a regular ring S. Then R is Cohen{Macaulay.
Proof. Already the proof of 6.5.2 depended on reduction to characteristic
p and eventually used a tight closure argument. However, not even in the
relatively `harmless' setting of 6.5.2 could the direct summand property be pushed through the reduction. Therefore we will have to prove a general local analogue of 6.5.4 from which we now derive the theorem. Being Cohen{Macaulay is a local property. Thus, let be a prime ideal of R . Then the hypotheses are inherited by the submodule R of S so that we may assume R is local with maximal ideal . Next we pass to the -adic completion R^ and the S -adic completion S^ of S . It is clear that R^ is a direct summand of S^ . Also regularity has survived. Indeed, one has S^ = S^ = S= S , and S^ is contained in the Jacobson radical of S^ (see 270], x8). It results from this fact and Nakayama's lemma that every maximal ideal of S^ is of the form S^ where is a maximal ideal of S . Now one uses the natural isomorphism between the -adic completion S1 of S and the S^ -adic completion S2 of S^ to conclude that S^ S^ is a regular local ring. The isomorphism of S1 and S2 is not hard to prove: choose systems of generators a1 . . . ar and b1 . . . bs of S and respectively with X = X1 . . . Xr and Y = Y1 . . . Ys one then has p
p
m
m
m
m
m
m
n
n
n
n
n
m
;
;
;
n
;
S1 = S X Y ]]=(X1 a1 . . . Xr ar Y1 b1 . . . Ys b1) S2 = (S X ]]=(X1 a1 . . . Xr ar ))Y ]]=(Y1 b1 . . . Ys bs )
;
;
;
;
p
407
10.4. Direct summands of regular rings
by 270], 8.12 (for the rst isomorphism we use that a1 . . . ar b1 . . . bs also generates ). From now on we may assume that R is a residue class ring of a Cohen{Macaulay ring. Notice that S is the direct product S1 Ss of regular integral domains. Let ei 2 S be the idempotent representing 1 2 Si . For the R -homomorphism : S ! R splitting the inclusion R ! S we have 1 = (e1 ) + + (es ). Since R is local, one of the (ei ) is a unit in R . It follows easily that the induced map R ! Si is split. Hence we can replace S by the domain Si especially, R is a domain. Now choose a system of parameters x1 . . . xd of R . Then n
((x1 . . . xm;1 ) :R xm )S = (x1 . . . xm;1)S
m = 1 . . . d
by 10.4.3 below. Moreover, IS \ R = I for all ideals I of R , and we conclude immediately that x1 . . . xd is an R -sequence, as desired. Remark 10.4.2. The theorem holds under the slightly weaker hypothesis that R is a pure subring of S (see 6.5.3(b) for this notion). In fact, purity implies that IS \ R = I for all ideals I of R furthermore it is stable under the reduction in the proof of 10.4.1 (see Hochster and Roberts 201], Section 6). If one assumes directly that R and S are domains and R is a residue class ring of a Cohen{Macaulay ring, then it is sucient that IS \ R = I for all ideals I of R (as was the case for 6.5.2 and 10.1.14). We refer the reader to 6.5.3 for a discussion of the predecessors and variants of 10.4.1. Theorem 10.4.3. Let (R ) be Noetherian local domain of dimension d that m
contains a eld and is a homomorphic image of a Cohen{Macaulay local ring. Furthermore let S be a regular domain extending R. Then one has ((x1 . . . xm;1) :R xm )S = (x1 . . . xm;1 )S for every system of parameters x1 . . . xd of R and m = 1 . . . d.
If the claim should fail, then there exists a maximal ideal of S such that ((x1 . . . xm;1 ) :R xm )S 6= (x1 . . . xm;1)S . Evidently = R \ must contain x1 . . . xm . In order to replace R by R we must only show that x1 . . . xm can be extended to a system of parameters of R . This holds if height(x1 . . . xm ) = m. Indeed, by assumption we have codim(x1 . . . xm ) = m, and R is a (universally) catenary local domain (see 2.1.12). In such a ring one has height I = codim I for all ideals I . After this rst step we can assume S is a regular local domain extending R . The completion of S (with respect to its maximal ideal) is a regular local ring extending S , and since it is faithfully at over S , there is no harm in supposing that S is even a complete regular local ring. The homomorphism R ! S induces a map R^ ! S^ = S , which however need not be injective. At least, its kernel is a prime ideal of R^ with Proof.
n
p
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n
p
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408
10. Tight closure
\ R = 0. Since R^ is at over R,
is a minimal prime ideal of R^ and ^ = dim R^ by virtue of Theorem 2.1.15. dim R= As in Section 8.1 we use the ideals i = Ann H i (R ). Set (R ) = 0 d ;1 and recall from 8.1.3 that (R ) ((x1 . . . xm;1) :R xm ) (x1 . . . xm;1)R furthermore, by 8.1.1, (R ) 6 . Since the image of (R ) under the map R ! S is non-zero, we can invoke the following lemma and conclude the proof. It is necessary to relax the condition that the homomorphism R ! S be injective. Actually we will have to reduce the next lemma to the case where this map is surjective in order to prove it in characteristic 0. (The hypothesis `complete' is only included to save us another reduction.) Lemma 10.4.4. Let (R ) be a complete Noetherian local ring containing a eld, (S ) a complete regular local ring, and ' : R ! S a ring homomorphism such that '( (R )) 6= 0. Then ((x1 . . . xm;1 ) :R xm )S = (x1 . . . xm;1)S for every system of parameters x1 . . . xd of R and m = 1 . . . d. Proof. Let us rst prove the lemma in characteristic p. Pick d 2 (R ) such that c = '(d ) 6= 0. For y 2 (x1 . . . xm;1) :R xm and all q = pe one has yq 2 (xq1 . . . xqm;1) :R xqm . Since xq1 . . . xqd is also a system of parameters, dyq 2 (xq1 . . . xqm;1). Applying ', we immediately see that '(y) 2 ((x1 . . . xm;1)S ) , whence '(y) 2 (x1 . . . xm;1)S by 10.1.7. The next step is the reduction to the case in which ' is surjective. By Cohen's structure theorem A.21 there are representations R = K Y1 . . . Yr ]]=I and S = LZ1 . . . Zs ]] where K = R= and L = S= are coecient elds of R and S , respectively. The map ' induces an inclusion K ! L so that we may view K as a sub eld of L. We set A = K Y1 . . . Yr ]] and A0 = LY1 . . . Yr ]]. Evidently, ' can only be surjective if K = L, and therefore we must extend R such that the extension R 0 has residue class eld L. Consider the homomorphism A ! S induced by '. It clearly factors through A0 . Therefore ' factors through R 0 = A0=IA0 . Note that R 0 is at over R : rst, it is easily proved that A0 is a at A-algebra (see Exercise 9.1.15), and, second, if C is an exact sequence of R -modules, then C R R 0 = C A A0 . 0 0 0 Moreover, R is the maximal ideal of R . Especially dim R 0 = dim R , and so every system of parameters of R is also a system of parameters of R 0 . We can replace R by R 0 if we have shown that (R )R 0 = (R 0). We set i = AnnA H i (R ), and de ne 0i similarly. Then i is the preimage of AnnR H i (R ), and the corresponding statement holds for 0i . Therefore q
q
q
a
a
m
a
a
a
a
q
a
m
n
a
a
m
n
.
.
m
.
m
a
b
b
a
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409
10.4. Direct summands of regular rings
it is enough that i A0 = 0i . By 8.1.1 we have i = AnnA M where M = ExtdA;i(R A). The atness of A0 over A implies M A0 = ExtdA;0 i(R 0 A0 ) 0 0 and AnnA (M )A = AnnA0 (M A ) (see the proof of 10.1.7). Using 8.1.1 once more, we arrive at the desired equality. From now on we may assume that K = L, R = R 0, and A = A0 . Next we extend ' to a surjection : Re ! S by choosing Re = R Z1 . . . Zs ]] and setting (Zj ) = Zj 2 S . The extension R ! Re is faithfully at, and every system of parameters of R can be extended by Z1 . . . Zs to a system of parameters of Re . Before we can replace R by Re , we need only to prove that (R )Re = (Re ). This however results again from 8.1.1 the reader can easily check that i Re = ei+s for all i = 0 . . . d and ei = Re for e = AZ1 . . . Zs]] and use that Re = R A Ae . i = 0 . . . s ; 1: set A We may now replace R by Re and A by Ae . After this change of notation the failure of the lemma can be described as follows: there exist b
a
b
b
a
a
a
a
(i) a regular local ring A with a regular system of parameters a1 . . . an , a residue class ring R = A=(b1 . . . bu) of dimension d , and a residue class ring S = A=(a1 . . . av ), v n, such that (b1 . . . bu ) (a1 . . . av ) (in fact, the kernel of the homomorphism A ! S is generated by a subset of a regular system of parameters) (ii) elements c0 . . . cd ;1 with ci 2 AnnA ExtdA;i (R A) and c = c0 cd ;1 2= (a1 . . . av )
(iii) elements x1 . . . xd 2 A whose residue classes form a system of parameters, a number m, 1 m d , and an element w 2 A such that wxm 2 (x1 . . . xm;1) + (b1 . . . bu), but xm 2= (x1 . . . xm;1 ) + (a1 . . . av ).
We want to show that, given such data in characteristic 0, we can also nd them in characteristic p. To this end we must show that the data above have a regular equational presentation. Theorem 8.4.1 then yields a characteristic p counterexample to our contention, the desired contradiction. All the relations `2' and `' can of course be expressed by polynomial equations. This holds also for the fact that b1 . . . bu and y1 . . . yd generate an ideal whose radical contains (a1 . . . an ). Furthermore we have already seen in 8.4.4 that the dimension condition in (i) and the non-membership relations in (ii) and (iii) can be captured by equations. For given i we write ExtdA;i(R A) as a residue class module As =W . Then the isomorphism ExtdA;i (R A) = As =W admits a regular equational presentation by 8.4.4, as does the relation ci As W for trivial reasons. This nally shows that all the data given in (i){(iii) can be encoded in a system of polynomial equations over Z.
410
10. Tight closure
Notes The fundamental paper for tight closure is Hochster and Huneke 192]. Essentially all of the material of Sections 10.1 and 10.2 has been taken from this source. A detailed discussion of the not yet solved localization and completion problems can be found in Huneke's lecture notes 215] which we have consulted extensively in writing Chapter 10. Much of the work that preceded tight closure theory and motivated its creation has been discussed Chapters 8 and 9. The theorem of Briancon and Skoda was originally proved by analytic methods, and the lack of an algebraic proof had been `for algebraists something of a scandal { perhaps even an insult { and certainly a challenge' (Lipman and Teissier 261]). As pointed out in Section 10.2 algebraic proofs of slightly di erent theorems were given by Lipman and Teissier and Lipman and Sathaye 260] the latter work uses di erential methods. The proof of the tight closure version by Hochster and Huneke is contained in their article 190], which is still very useful as a rst overview of our subject. For variants and generalizations of the Briancon{ Skoda theorem see Aberbach and Huneke 3], 4], and Swanson 368]. For the connection with reduction numbers and Rees algebras see Aberbach, Huneke, and Trung 6]. F -rational (local) rings appeared rst in Fedder and Watanabe 104]. Our treatment of their basic properties essentially follows Huneke 215]. The connection with local cohomology and the Frobenius action on it goes back to the work 202] of Hochster and Roberts that introduced F -purity. Special cases of Smith's theorem 349] that F -rational type implies rational singularity and its converse by Hara 149], which we have quoted in 10.3.27, had been proved in special cases by Fedder 101], 102], 103] and Hochster and Huneke 190]. The proof of 10.3.24 was suggested to us by Watanabe. We could only prove the easiest result on the existence of test elements that in its general version presents the perhaps most intricate aspect of tight closure see Hochster and Huneke 192], 195]. The existence of test elements is closely related with the so-called persistence theorem that under suitable conditions guarantees the relation '(I ) ('(I )S ) for a ring homomorphism ' : R ! S see 195], (6.24). Some results about the hierarchy of `F -properties' have been indicated in Section 10.3. For more information, especially for examples delimiting these properties from each other and for the relation to singularity theory, the reader is referred to Fedder and Watanabe 104], Watanabe 390], 391], and Hara and Watanabe 150]. Theorem 10.4.1 was stated by Hochster and Huneke 190], 3.5 without proof. A complete proof appeared in their paper 197]. It uses the functoriality of big Cohen{Macaulay algebras. Our derivation of the
Notes
411
theorem is certainly a variant of the idea behind 190], 3.5. The de nition of tight closure can be extended from the situation I R to that in which U is a submodule of the R -module M . In particular, this leads one to the notion of phantom homology and phantom acyclicity see Hochster and Huneke 192], 194]. For phantom acyclic complexes one has a vanishing theorem similar to 10.4.3 where the ideal quotient is replaced by the homology of a phantom acyclic complex. It seems however that the strongest such vanishing theorem 197], (4.1) needs big Cohen{Macaulay algebras. One can also derive a `phantom' version of the `improved new intersection theorem' 9.4.2 thus tight closure o ers another approach to the homological theorems of Chapter 9. Aberbach has developed `phantom' homological algebra that includes phantom projective dimension, phantom depth and an Auslander{Buchsbaum formula. Tight closure can be also de ned in characteristic 0 see 198] and the Appendix of 215] by Hochster. So far there seems to be no de nition of tight closure in mixed characteristic. Hochster has developed a theory of solid closure that does not depend on characteristic 188]. For `good' rings of characteristic p > 0 solid closure coincides with tight closure however, there exist examples showing that ideals in a regular ring containing a eld of characteristic 0 need not be solidly closed. There are many more aspects and applications of tight closure. We content ourselves with a list of cues and references: tight closure in graded rings (Smith 350]), Hilbert{Kunz functions and multiplicities (Kunz 248], Monsky 277]), uniform Artin{Rees theorems (O'Carroll 292], Huneke 214]), arithmetic Macaulay cation (Huneke and Smith 5]), strongly F -regular rings (Hochster and Huneke 191], Glassbrenner 127]), di erentially simple rings (Smith and Van den Bergh 352]), Kodaira vanishing and other vanishing theorems of algebraic geometry (Huneke and Smith 218], Smith 351]).
Appendix: A summary of dimension theory
Dimension theory is a cornerstone of commutative ring theory, and is covered by every serious introduction to the subject. For ease of reference we have collected its main theorems in this appendix, together with the structure theorems for complete local rings. Most of the theorems below have the names of their creators associated with them and should be easily located in the literature. For some of the results we outline a proof. Height and dimension. There exist two principal lines of development for general dimension theory. The rst and `classical' approach, to which we shall adhere, starts from the Krull principal ideal theorem (47], 231], 284], 344], 397]) whereas the second brings the Hilbert{Samuel function into play at a very early stage (15], 270]). Let R be a commutative ring, and 2 Spec R . The height of is the supremum of the lengths t of strictly descending chains = 0 1 t of prime ideals. For an arbitrary ideal I one sets height I = inf fheight : 2 Spec R I g: The fundamental theorem on height is Krull's principal ideal theorem: Theorem A.1. Let R be a Noetherian ring, and I = (x1 . . . xn ) a proper ideal. Then height n for every prime ideal which is minimal among p
p
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p
p
p
p
p
p
p
p
the prime ideals containing I.
In particular, every proper ideal in a Noetherian ring has nite height. In a sense, the following theorem is a converse of the principal ideal theorem. Theorem A.2. Let R be a Noetherian ring, and I a proper ideal of height n. Then there exist x1 . . . xn 2 I such that height(x1 . . . xi ) = i for i = 1 . . . n. The elements xi are chosen successively such that xi is not contained in any minimal prime overideal of (x1 . . . xi;1) that such a choice is possible follows from 1.2.2. 412
413
A summary of dimension theory
The (Krull) dimension of a ring R is the supremum of the heights of its prime ideals, dim R = supfheight : 2 Spec R g: Because of the correspondence between Spec R , 2 Spec R , and the set of prime ideals contained in , one has dim R = height : A fundamental and very easily proved inequality is height I + dim R=I dim R for all proper ideals I of R . The dimension of a Noetherian local ring can be characterized in several ways: Theorem A.3. Let (R ) be a Noetherian local ring, and n 2 N. Then the p
p
p
p
p
p
p
m
following are equivalent: (a) dim R = n (b) height = n (c) n is the in mum of all m for which there exist x1 . . . xm with Rad(x1 . . . xm) = . (d) n is the in mum of all m for which there exist x1 . . . xm such that R=(x1 . . . xm ) is Artinian.
2
m
2
m
m
m
The equivalence of (a) and (b) is trivial. That of (b) and (c) results from A.1 and A.2, and for (c) () (d) one uses the fact that a Noetherian ring is Artinian if and only if all its prime ideals are maximal, in other words, if it has dimension 0. If dim(R=(x1 . . . xn)) = 0 with n = dim R , then x1 . . . xn is a system of parameters of R . Sometimes it is appropriate to use the codimension of an ideal in a ring R which is given by codim I = dim R ; dim R=I: Dimension of modules. The notion of dimension can be transferred to modules. Let M be an R -module then dim M is the supremum over the lengths t of strictly descending chains
= 0 1 t with i 2 Supp M: In the case of main interest in which M is a nite module one has Supp M = f 2 Spec R : Ann M g so that dim M = dim(R= Ann M ). If (R ) is local, then a system of parameters for a non-zero nite R -module M is a sequence x1 . . . xn 2 , n = dim M , such that M=(x1 . . . xn)M is Artinian. The following inequality is often useful: p
p
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414
Appendix
Proposition A.4. Let (R ) be a Noetherian local ring, M a nite R-module, and x1 . . . xr 2 . Then dim(M=(x1 . . . xr )M ) dim M ; r equality holding if and only if x1 . . . xr is part of a system of parameters m
m
of M.
This is easy. One rst replaces M by R= Ann M so that it is harmless to assume M = R . Then one chooses y1 . . . ym 2 such that their residue classes in R=(x1 . . . xr ) form a system of parameters. Finally one applies A.3. An important datum of a homomorphism of local rings (R ) ! (S ) is its bre S= S . For example it relates the dimensions of R and S : Theorem A.5. Let (R ) ! (S ) be a homomorphism of Noetherian local m
m
n
m
m
n
rings. (a) Then dim S dim R + dim S= S (b) more generally, if M is a nite R-module and N is a nite S-module, then dimS (M R N ) dimR M + dimS N= N.
m
For the proof of (b) set I = Ann M and R = R=I . Then U R N = Thus we may replace R by R , S by S=IS , and N by N=IN . That is, we may assume Supp M = Spec R . Next, replacing S by S=(Ann N ), we may suppose Supp N = Spec S . Under these conditions the desired inequality is equivalent with (a). For (a) one chooses a system x1 . . . xn of parameters of R , and uses that Rad(x1 . . . xn)S = Rad S . U
m
R N=IN for every R -module U.
m
Integral extensions. Recall that an extension R S of commutative rings is integral if every element x 2 S satis es an equation xn + an;1xn;1 + + a0 = 0 with coecients ai 2 R. Very often one uses that S is a nite R -module if and only if it is an integral extension and nitely generated as an R -algebra. Theorem A.6. Let R S be an integral extension, and 2 Spec R. (a) There exists a prime ideal 2 Spec S with = \ R (one says lies over ) (b) there are no inclusions between prime ideals lying over (c) in particular, when lies over , then is maximal if and only if is. The following theorem comprises the Cohen{Seidenberg going-up and going-down theorems. Theorem A.7. Let R S be an integral extension. (a) If 0 are prime ideals of R and 2 Spec S lies over , then there p
q
p
q
q
p
p
q
exists a prime ideal 0 p
p
q
p
p
q
in S lying over 0 q
q
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415
A summary of dimension theory
(b) if, in addition, S is an integral domain and R is integrally closed, then,
given prime ideals 0 of R and of S, lying over , there exists 0 Spec S, 0 , which lies over 0 . Corollary A.8. Let R S be an integral extension of Noetherian rings, and I a proper ideal of S. Then dim S=I = dim R=(I R ). The rst step in proving the corollary is to replace S by S=I and R by R=(I R ) so that one may assume I = 0. Then given a strictly descending chain 0 1 t of prime ideals, the chain of prime ideals R i is also strictly descending by A.6, and conversely, given a chain in Spec R , one constructs a chain of the same length in Spec S using A.7(a). In general, one says that going-up or going-down holds for a ring homomorphism R S if it satis es mutatis mutandis the conclusions of q
2
q
\
p
q
q
q
p
q
q
p
p
\
\
q
q
!
A.7(a) or (b) respectively.
Flat extensions. It is an important fact that atness implies going-down: Lemma A.9. Let R S be a homomorphism of Noetherian rings, and suppose there exists an R-at nite S-module N with Supp N = Spec S. Then going-down holds.
!
Going-down can be reformulated as follows: for all prime ideals 2 Spec R and 2 Spec S lying over the natural map Spec S ! Spec R is surjective. Now, given such prime ideals and , N is even a faithfully at R -module, and the surjectivity of Spec S ! Spec R follows from the next lemma. Lemma A.10. Let R ! S be a ring homomorphism. If an S-module N is faithfully at over R, then the associated map Supp N ! Spec R is p
q
p
q
p
p
q
q
p
q
p
surjective.
In fact, let 2 Spec R we set k( ) = R = R . Then k( ) R N 6= 0, and the support of the k( ) R S -module k( ) R N contains a prime ideal . If we choose = S \ , then 2 Supp N , and furthermore \ R = : one has \ R = \ R, and \ R = since the map Spec(k( ) R S ) ! Spec R factors through Spec k( ). For at extensions the inequalities in A.5 become equations: Theorem A.11. Let (R ) ! (S ) be a homomorphism of Noetherian p
p
p
p
Q
p
q
q
p
Q
q
p
p
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q
Q
Q
p
p
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m
n
local rings. (a) If S is a at R-algebra, then dim S = dim R + dim S= S (b) more generally, if M is a nite R-module and N is an R-at nite S-module, then dimS (M R N ) = dimR M + dimS N= N. m
m
As we did for A.5, the theorem is easily reduced to the case in which Supp N = Spec S . By virtue of the previous lemma the homomorphism R ! S then satis es going-down. We choose a prime ideal of S which contains S and has the same height as S . Then going-down q
m
m
416
Appendix
immediately implies height
height
q
m
. Hence
dim S height S + dim S= S height + dim S= S m
m
m
m
as desired. The converse inequality is part of A.5. Polynomial and power series extensions. The dimension of a polynomial or power series extension is easily computed: Theorem A.12. Let R be a Noetherian ring. Then
dim R X ] = dim R X ]] = dim R + 1: Let S = R X ] or S = R X ]]. Then R = S=(X ), and since height(X ) = 1 one has dim S dim R + 1. For the converse we rst consider the polynomial case. Let be a maximal ideal of R X ], and set = R \ . As S = R X ] is R -at, one may apply A.11, and only needs to show that dim(S = S ) = 1. It is a routine matter to check that S = S is a localization of the polynomial ring (R = R )X ] with respect to a maximal ideal. Since R = R is a eld, S = S is a discrete valuation ring and therefore of dimension 1. In the power series case is always a maximal(!) ideal of R , and S = S is therefore the discrete valuation ring (R= )X ]]. Corollary A.13. Let k be a eld. Then n
p
n
p
n
p
n
n
n
p
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p
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n
n
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p
dim kX1 . . . Xn] = dim kX1 . . . Xn]] = n: Ane algebras. Let k be a eld. A nitely generated k-algebra R is called an ane k-algebra. Excellent sources for the theory of ane algebras are
Kunz 249] and Sharp 344]. The key result is Noether's normalization theorem : Theorem A.14. Let R be an ane algebra over a eld k, and let I be a proper ideal of R. Then there exist y1 . . . yn 2 R such that (a) y1 . . . yn are algebraically independent over k (b) R is an integral extension of ky1 . . . yn ] (and thus a nite ky1 . . . yn]module) (c)
\ ky1 . . . yn] = X yiky1 . . . yn] = (yd+1 . . . yn) i=d +1 for some d, 0 d n. I
n
Moreover, if y1 . . . yn satisfy (a) and (b), then n = dim R.
417
A summary of dimension theory
If y1 . . . yn satisfy (a) and (b), then ky1 . . . yn] is called a Noether normalization of R . That necessarily n = dim R follows from A.8 and A.13. That condition (c) can be satis ed in addition to (a) and (b) is crucial for dimension theory. The graded variant of Noether normalization (due to Hilbert) is given in 1.5.17. An important consequence of Noether normalization is (the abstract version of) Hilbert's Nullstellensatz : Theorem A.15. Let k be a eld, and K an extension eld of k which is a
nitely generated k-algebra. Then K is a nite algebraic extension of k. In fact, if ky1 . . . yn] is a Noether normalization of K , then n = dim K = 0 and K is an integral extension of k, from which one easily
concludes that it is a nite algebraic extension. The following theorem contains the main results of the dimension theory of ane algebras. Theorem A.16. Let R be an ane algebra over a eld k. Suppose that R
is an integral domain. Then (a) dim R = tr degk Q(R ) where tr degk Q(R ) is the transcendence degree of the eld of fractions of R over k, (b) height = dim R dim R= for all prime ideals of R. For part (a) we choose a Noether normalization ky1 . . . yn]. Then Q(R ) is algebraic over Q(ky1 . . . yn]), and the latter has transcendence degree n over k. For part (b) we require in addition that ky1 . . . yn] satis es A.14 for I = . Then the image of kX1 . . . Xd ] in R= is a Noether normalization for that ring, whence dim R= = d . On the other
;
p
p
p
p
p
hand, note that going-down holds according to A.7: being a factorial ring (a UFD in other terminology) ky1 . . . yn] is integrally closed. It follows that height height \ ky1 . . . yn] = n ; d . Summing up, we have height + dim R= n = dim R , and the converse inequality is automatic as noticed above. p
p
p
p
p
Hilbert rings. It is a consequence of Hilbert's Nullstellensatz that a prime
ideal in an ane algebra over a eld is the intersection of the maximal ideals in which it is contained. Rings with this property are therefore called Hilbert rings (Bourbaki prefers the term Jacobson rings). The following is the main theorem on Hilbert rings: Theorem A.17. Let R be a Hilbert ring, and S a nitely generated R-
algebra. Then (a) S is a Hilbert ring, (b) R is a maximal ideal of R for every maximal ideal of S. Corollary A.18. Let R be a nitely generated Z-algebra, and a maximal ideal of R. Then Z = (p) for some prime number p Z, and R= is a nite eld. m
\
m
m
\
2
m
m
418
Appendix
In fact, Z is a Hilbert ring, and R= is a nite algebraic extension of Z=(p) by A.15. m
A dimension inequality. For the study of the dimension of Rees rings
and associated graded rings the following theorem (due to Cohen) is important. Theorem A.19. Let R S be an extension of integral domains, and suppose R is Noetherian. Let 2 Spec S and = \ R. Then dim S + tr degQ(R= ) Q(S= ) dim R + tr degQ(R) Q(S ): P
p
P
P
P
p
p
We reproduce the proof given in 270], x15. The rst step is a reduction to the case in which S is a nitely generated R -algebra. There is nothing to prove if the right hand side is in nite. So suppose it is nite, and let m and t be integers with 0 m dim S , 0 t tr degQ(R= ) Q(S= ). Then there exists a strictly descending chain = 0 m of prime ideals in S . We choose ai+1 2 i n i+1, and furthermore elements c1 . . . ct 2 S whose residue classes in S= are algebraically independent over Q(R= ). Let S 0 = R a1 . . . am c1 . . . ct ], and 0 = \ S 0 then dim S 0 0 m and tr degQ(R= ) Q(S= 0 ) = t. Thus it is enough to prove the claim for S 0 and C 0. In the case in which S is nitely generated, we use induction on the number of generators so that only the case S = R x] remains. Write S = R X ]= . If = 0, then S = R X ], and dim S = dim R + dim(S = S ) by A.11. As S = S is a localization of Q(R= )X ], we have dim(S = S ) = 1 ; tr degQ(R= ) Q(S= ) = tr degQ(R) Q(S ) ; tr degQ(R= ) Q(S= ). In the case 6= 0 we have tr degQ(R) Q(S ) = 0. Since R is a subring of S , \ R = 0 so that R X ] is a localization of Q(R )X ], and therefore has dimension 1, equivalently height = 1. Let 0 the inverse image of in R X ], and note that Q(R X ]= 0 ) = Q(S= ) in a natural way. Then dim S dim R X ] 0 ; height = dim R + 1 ; tr degQ(R= ) Q(R X ]= 0 ) ; 1 = dim R ; tr degQ(R= ) Q(S= ): P
P
p
P
P
P
P
P
P
p
P
P
P
p
P
Q
Q
p
P
p
p
P
P
p
P
p
P
P
P
P
P
p
p
Q
Q
Q
Q
P
P
P
P
Q
P
P
P
p
p
P
p
p
Complete local rings. The theory of Noetherian complete local rings, for which we recommend Matsumura 270] or Bourbaki 47] as a source, leads to similar results as that of ane algebras. For the relation between the characteristic char R of a local ring (R ) and that of its residue eld R= one of the following cases holds true: (i) char R= = 0 then R contains the eld Q of rational numbers, in particular char R = 0 (ii) char R= = p > 0 and char R = p too then m
m
m
m
419
A summary of dimension theory
R contains the eld Z=pZ (iii) char R= = p > 0 and char R = 0 (the typical case in number theory) (iv) char R= = p > 0 and char R = pm for some m > 1. In cases (i) and (ii) one says that R is equicharacteristic. (Note that R does not contain a eld in cases (iii) and (iv), and that (iv) m
m
is excluded for a reduced ring.) Theorem A.20. Let (R ) be a Noetherian complete local ring. (a) If R is equicharacteristic, then it contains a coecient eld, i.e. a eld m
k which is mapped isomorphically onto R= by the natural homomorphism R R= . (b) Otherwise let p = char R= . Then there exists a discrete valuation ring (S pS ) and a homomorphism ' : S R which induces an isomorphism S=pS R= and furthermore (i) is injective, if char R = 0, (ii) has kernel pm S, if char R = pm . It is a standard technique to pass from a Noetherian local ring (R ) to its completion R^ (with respect to the -adic topology). Then one is in
!
m
m
m
!
!
m
m
a position to apply Cohen's structure theorem : Theorem A.21. Let (R ) be a Noetherian complete local ring. Then there m
m
exists a ring R0 which is a eld or a discrete valuation ring such that R is a residue class ring of a formal power series ring R0 X1 . . . Xn]]. In fact, let x1 . . . xn be a system of generators of . Then there exists a uniquely determined homomorphism ' : R0X1 . . . Xn ]] R with '(Xi) = xi where R0 is either a coecient eld of R or, in the case of unequal characteristic, a discrete valuation ring S according to A.20. In Section 2.2 it is shown that R0X1 . . . Xn]] is a regular local ring, and m
!
often one uses A.21 `only' to the extent that a complete local ring is a residue class ring of a regular local ring. The analogue of Noether normalization is Theorem A.22. Let (R ) be a Noetherian complete local ring, and suppose m
that R is equicharacteristic or a domain. (i) In the equicharacteristic case let R0 R be a coecient eld of R, and y1 . . . yn a system of parameters (ii) otherwise, let p = char R= and R0 R be a discrete valuation ring according to A.20, and y1 . . . yn be elements such that p y1 . . . yn is a system of parameters. Then R is a nite R0y1 . . . yn]]-module, and R0y1 . . . yn ]] is isomorphic to the formal power series ring R0 Y1 . . . Yn ]]. One rst shows that R is a nite R0 y1 . . . yn ]]-module, so that dim R = dim R0y1 . . . yn]]. The substitution Yi yi induces a surjective homomorphism ' : R0 Y1 . . . Yn ]] R0 y1 . . . yn ]] which is also injective since dim R0Y1 . . . Yn ]] = dim R0y1 . . . yn ]].
m
!
7!
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Notation
0^ 1^ least, greatest element of a poset, 215
a1 . . . am ] maximal minor of a matrix, 306
a1 . . . au j b1 . . . bu ] minor of a matrix, 306 ad 161 ad 162 a X a ne hull of the set X , 224 a(R ) a-invariant of the graded ring R , 140, 176 A0 ( R ) category of Artinian graded R -modules, 142 A(R) category of Artinian R -modules, 105 i(M ) i-th Betti number of the module M , 16 C eld of complex numbers C. 130 Ce() Ce (; ) augmented oriented chain complex of the simplicial complex , the cell complex ; , 229, 266 char k characteristic of the eld (ring) k () Euler characteristic of the simplicial complex , 212 e () reduced Euler characteristic of , 231 j (x M ) j -th partial Euler characteristic of H. (x M ), 197 IM (n) Hilbert{Samuel function of M with respect to I , 188 (x M ) Euler characteristic of H. (x M ), 193 Cl (R ) divisor class group of the ring R C (n d ) cyclic polytope, 226 cn () cone over the simplicial complex , 230 codim F. codimension of the complex F., 352 codim I codimension of the ideal I , 413 conv X convex hull of the set X , 223 core core of the simplicial complex , 244 deg x degree of the element x, 28 d (d times iterated) di erence operator, 148 312 (P ) vertex scheme of the polytope P , 225 ( ) order complex of the poset , 208 r r-skeleton of the simplicial complex , 221 depth M depth of the module M , 10 det 1 inverse determinant character, 279 df di erential of the Koszul complex of f , 43 dfM di erential of K. (f M ), 44 dim M Krull dimension of the module M , 413 dim R dimension of the graded ring R , 35 h i
h i
;
439
440
Notation
dim R E (M ) E (M ) embdim R e(I M ) e(M ) e(x M ) "1 (R ) "2 (R ) Exti (M N )
F F(D) F(P ) f () F0 ( R ) F(R) ;
; ; (M ) G GL (n k) GL (V ) grade I grade(I M ) grade M grF (R ) grF (M ) grI (M ) grI (R ) g? G(X ) GB (X ) H+ H height I H . (f ) H. (f ) H . (f M ) H. (f M ) e i ( G) H e i ( G) H Hi (M n) H (M n) HM (t) H i (M ) H i (M ) H . (F . ) H. (F. ) H i (F . ) m
;
m
m
Krull dimension of the ring R , 413 injective hull of the R -module M , 98 injective hull of the graded module M , 137 embedding dimension of the local ring R , 74 multiplicity of M with respect to the ideal I , 188 multiplicity of the (graded) module M , 150, 188 multiplicity symbol, 193 rst deviation of the local ring R , 75 second deviation of the local ring R , 81 i-th graded extension module of M by N , 33 Frobenius functor, 328 face lattice of the cone D, 262 face lattice of the polyhedron P , 224 f -vector of the simplicial complex , 208 category of nite graded R -modules, 142 category of nite R -modules, 105 join of the simplicial complexes ; and , 221 312 submodule of elements of M with support in f g, 127 residue class ring of G(X ), 312 group of invertible n n matrices over k group of automorphisms of the vector space V grade of the ideal I , 12 grade of the ideal I with respect to the module M , 10 grade of the module M , 12 associated graded ring with respect to ltration F , 182 182 associated graded module of M with respect to ideal I , 6 associated graded ring of R with respect to ideal I , 6 initial form of g, 186 B -algebra generated by the maximal minors of X , 306 closed half-spaces de ned by the hyperplane H , 223 height of the ideal I , 412 cohomology of the Koszul complex of f , 45 homology of the Koszul complex of f , 44 cohomology of K . (f M ), 45 homology of K. (f M ), 44 i-th reduced simplicial cohomology of with values in G, 230 i-th reduced simplicial homology of with values in G, 230 i-th iterated Hilbert function of the module M , 150 Hilbert function of the graded module M , 147 Hilbert series of the graded module M , 147 i-th local cohomology module of the module M , 143 i-th local cohomology module of the module M , 128 cohomology of the complex F . homology of the complex F. i-th cohomology of the complex F .
m
441
Notation Hi (F. )
HomR (M N ) H. (R ) Hi (R )j
I ] I
I
idM inj dim M inj dim M
I q ] I I I? It (') It (U ) k C ] k C ]F k ] K . (f ) K. (f ) K . (f M ) K. (f M ) K. (x) k( ) (I ) (I M ) `(M ) L(f ) L(I ) lim Mi lim Mi lk F
p
;! ;
log
Lu M0(R)
M M Mf ^ M M (i) M (t) MG(t) M( ) MS
p
i ( M )
(M ) p
i-th homology of the complex F.
group of homogeneous homomorphisms ' : M ! N , 33 Koszul algebra of the local ring R , 75 80 divisor class of the ideal I integral closure of the ideal I , 388 312 identity map on the set M injective dimension of the graded module M , 138 injective dimension of the module M , 92 q-th Frobenius power of I , 328 ideal generated by the homogeneous elements in I , 28 tight closure of the ideal I , 378 ideal generated by leading monomials of the elements of I , 157 ideal generated by the t-minors of (a matrix of) ', 21 ideal generated by the t-minors of the matrix U , 21 a ne semigroup ring over C , 256 262 Stanley{Reisner ring of the simplicial complex , 209 dual of the Koszul complex with respect to f , 45 Koszul complex of f , 44 dual of K. (f ) with respect to M , 45 Koszul complex of f with coe cients in M , 44 Koszul complex with respect to sequence x, 46 residue class eld of localization with respect to , 9 analytic spread of the ideal I , 190 analytic spread of the ideal I with respect to M , 190 length of the module M leading monomial of f , 157 set of the leading monomials of the elements of I , 157 direct limit of the modules Mi inverse limit of the modules Mi link of the face F with respect to , 232 257 158 category of graded R -modules, 28 (bi)dual of the module M , 19 module of fractions with respect to the powers of f completion of the module M over a local ring module with shifted grading, 32 Molien series of the character , 284 Molien series of the group G, 283 homogeneous localization of M with respect to , 31 module of fractions of M with respect to the multiplicatively closed set S i-th Bass number of M with respect to , 101 minimal number of generators of M , 16
p
p
p
442
N
nat
!R (x) Pf (') pf (') PM (n)
O
proj dim M
Q Q+ QC
rank M rank ' rank(' M ) rank v R
R
R R+ R+S RC
R+(F ) R+(F M) R(F ) R(F M) R(I M) reg(M ) relint C relint X
Rf RG R^ rk () R M (R ) (R ) (R k )
m
m
m
r(M ) (Rn ) R( ) R 1=q Rr+1 Rr+1(X ) RS p
Ru
Notation
set of non-negative integers natural map canonical module of the ring R , 110 order ideal of the element x of a module, 360 ideal generated by the submaximal Pfa ans of ', 121 Pfa an of the matrix ', 120 Hilbert polynomial of the graded module M , 149 projective dimension of the module M , 16 eld of rational numbers set of non-negative rational numbers Q-vector space generated by the a ne semigroup C , 257 rank of the module M , 20 rank of the homomorphism ', 20 rank of ' with respect to the module M , 351 rank of the poset element v, 215 module of semi-invariants of the ring R with respect to the character , 279 determinantal ring, 312 eld of real numbers set of non-negative real numbers cone generated by the set S , 258 R-vector space generated by the a ne semigroup C , 257 Rees ring with respect to the ltration F , 182 182 extended Rees ring with respect to the ltration F , 182 182 182 regularity of the graded module M , 168 relative interior of the a ne semigroup C , 261 relative interior of the set X , 225 ring of fractions with respect to the set of powers of f ring of invariants of R under the action of G, 278 completion of the local ring R type of the simplicial complex over the eld k, 240 trivial extension of the ring R by the module M , 111 local ring R with maximal ideal , 35 local ring R with maximal ideal local ring R with maximal ideal and residue class eld k = R= type of the module M , 13 Serre's condition (Rn ), 71 homogeneous localization of R with respect to , 31 ring of q-th roots of the elements of R , 398 determinantal ring, 306 ring of fractions of R with respect to the multiplicatively closed set S 158
m
m
m
m
p
Notation R
X1 . . . Xn ]] R
Sing R SL (n k) SL (V ) MI (n) (Sn ) Soc M Soc M st F supp a supp u supp xa S (V ) tr degk K Tr
u v V (I ) vol P Vi M V VM ' xF x y xy
^ h i
Z ZC
443 formal power series ring over R set of elements of R not contained in a minimal prime ideal, 378 singular locus of R , 382 group of n n matrices over k with determinant 1 group of automorphisms of V with determinant 1 Hilbert{Samuel polynomial of M with respect to I , 188 Serre's condition (Sn ), 63 socle of the module M , 15 homogeneous socle of the graded module M , 142 star of the face F with respect to , 232 support of the element a 2 Zn , 211 support of the monomial u, 300 support of the monomial xa , 211 symmetric algebra of V , 278 transcendence degree of K over k trace of the linear map , 284 v covers u, 215 set of prime ideals containing I volume of the polytope P , 277 i-th exterior power of the module M , 41 exterior algebra of the module M , 40 extension of ' to exterior algebra, 41 monomial corresponding to the face F , 220 product of x and y in exterior algebra, 40 scalar product of x and y, 223 ring of integers group generated by the a ne semigroup C , 256
Index
absolute integral closure, 382 acyclic complex, 23 acyclicity criterion, 24, 25, 27, 351 admissible grading, 260 a ne algebra, 416{417 a ne hull, 224 a ne semigroup, 256 simplicial, 269 a ne semigroup ring, 256{263, 305 and regularity, 263 Cohen{Macaulay property of, 269 graded prime ideals of, 261 local cohomology of, 266{269 a nely (in)dependent, 225 a-invariant free resolution and, 141 of a Veronese subring, 144 of a Cohen{Macaulay ring, 141 of a Gorenstein ring, 140 of a positively graded algebra, 176 of a simplicial complex, 222 a-invariant and F -rationality, 405 Alexander duality, 241 algebra structure on a resolution, 121 algebra with straightening law, see graded ASL alternating graded algebra, 40 alternating homomorphism, 120 analytic deviation, 202 analytic spread, 190 inequalities for, 191 analytically independent, 190 antiderivation, 43 Artin's approximation theorem, 340 Artinian, 142 ASL, see graded ASL associated graded module, 6, 7, 182
associated graded ring, 6, 182, 303 dimension of an, 185 augmented oriented chain complex, 229, 266 Auslander's conjecture, 365 Auslander{Buchsbaum formula, 17 Auslander{Buchsbaum{Nagata theorem, 70 Auslander{Buchsbaum{Serre theorem, 66 Bass number, 101{103, 135, 138, 372{375 Bass' conjecture, 96, 374 Betti number, 16, 17, 367{371 and characteristic, 243 ne, 240 graded, 37 bidual, 19 big Cohen{Macaulay algebra, 347, 382 big Cohen{Macaulay module, 323, 334, 336 balanced, 342{346, 352 completion of, 343 big rank, 360 boundary complex, 224 Boutot's theorem, 293 Brian!con{Skoda theorem, 391{393 Bruggesser{Mani theorem, 228 Buchberger algorithm, 158 Buchsbaum ring, module, 86 Buchsbaum{Eisenbud acyclicity criterion, 23{25, 351 structure theorem, 121
canonical element theorem, 359, 361, 366
444
445
Index
canonical module, 107{120 and nite extension, 112 and at extension, 115{116, 118 and ground eld extension, 120 and polynomial extension, 118 and power series extension, 118 class of, 315 completion of, 110 existence of, 111, 113 free resolution of, 113 is divisorial ideal, 117 localization of, 110 modulo regular sequence, 110, 118 of a graded ring, 136{144 of a non{Cohen{Macaulay ring, 134 of an Artinian algebra, 113 rank of, 117 uniqueness of, 109 canonical module, 139{144 existence of, 140, 143 localization of, 140 modulo regular sequence, 141 uniqueness of, 139 catenary, 62 ' complex Cech modi ed, 130 cell complex, 264{266 chain, 208 character, 279 inverse determinant, 279 class group, 315 of a localization, 315 of a polynomial extension, 315 of a positively graded algebra, 315 closed half-space, 223 codimension conjecture, 366 of a complex, 352, 361, 368, 370 of an ideal, 413 coe cient eld, 419 Cohen's structure theorem, 419 Cohen{Macaulay approximation, 119 Cohen{Macaulay complex, 210, 219, 346 and base ring, 222 and links, 237
doubly, 252{254 level, 240, 252 topological characterization of, 239 type of, 240 upper bound for h-vector, 213 Cohen{Macaulay module, 57{65, 68, 69 and completion, 60 and at extension, 60 and polynomial extension, 60 and power series extension, 60 big, see big Cohen{Macaulay module graded, 64 localization of, 58 maximal, 57, 73, 108 modulo regular sequence, 58 multiplicity of, 197 Cohen{Macaulay ring, 57{65, 68, 96, 367, 374 see also Cohen{Macaulay module and F -rationality, 394 and associated graded ring, 186 and base eld extension, 61 and faithfully at extension, 64 and Reynolds operator, 282 graded, 64 is universally catenary, 62 one dimensional, 192 unmixedness of ideals in, 59 Cohen{Seidenberg theorems, 414 coideal in a poset, 312 complete, 142 complete intersection, 73{85, 96, 119, 367 and at extension, 77 and ground eld extension, 77 and polynomial extension, 77 and power series extension, 77 graded, 85 localization of, 76, 77 locally, 77{78, 85 modulo regular sequence, 76 complete intersection defect, 77 complete local ring, 62, 113, 133, 418{419 completely integral, 186
446 completely integrally closed, 186 complex with nite length homology, 324, 327, 329, 362 cone, 230, 258 convex combination, 223 hull, 223 set, 223 cotangent module, 200 cover, 215 cross-section, 262 cyclic polytope, 226{227 decomposable module, 100 Dehn{Sommerville equations for Euler complexes, 238 for polytopes, 228, 229 dehomogenization, 38{39, 73 depth, 8{16, 96, 353 and dimension, 12 and Ext, 11 and nite extension, 15 and at extension, 13 and projective dimension, 16{18 of a non- nite module, 350 of completion, 60 determinantal ring, 306, 311{318 a-invariant of, 317 canonical module of, 316 Cohen{Macaulay property of, 312 dimension of, 312 Gorenstein property of, 316 normality of, 312 of alternating matrix, 318 of symmetric matrix, 317, 318 Serre condition (R2 ) for, 315 diagonalizable group, 279 di erence operator, 148 dimension, 304, 305, 365, 412{418 and at extension, 415{416 and polynomial extension, 416 and power series extension, 416 inequality, 418 of completion, 60 dimension, 35 direct summand theorem, 356, 361 divisible module, 91 divisor class group, see class group
Index
divisorial ideal, 315 doset algebra, 318 d -sequence, 202 dual, 19 edge, 208 Ehrhart function, series, 275 quasi-polynomial, 276 embedded deformation, 200 embedding dimension, 74 equicharacteristic local ring, 419 equidimensional, 383 equimultiple ideal, 202 essential extension, 97 maximal, 98 proper, 97 essential extension, 136 proper, 136 Euler characteristic and multiplicity, 195 and multiplicity symbol, 193 of a simplicial complex, 212, 231 reduced, 231 of the Koszul homology, 193 partial, 197 Euler complex, 238 f -vector of, 239 Euler relation, 229 Evans{Gri th syzygy theorem, 371 excellent ring, 382 expanded subsemigroup, 290 expected rank, 24, 351, 353, 370 exterior algebra, 40{43, 79, 80 of free module, 42 of graded module, 43 exterior power, 41, 53{54
face lattice, 224, 262 face of a cell, 264 face of a simplicial complex, 207 dimension of, 207 face ring, 209 facet of a simplicial complex, 208 factorial ring, 70{72, 117, 315 F - nite ring, 398 bre, 414 ltered ring, 182
Index
ltration, 182 Noetherian, 183, 184 separated, 183 strongly separated, 183 ne grading, 210 nite at dimension, 87 nite regular cell complex, see cell complex F -injective ring, 401 rst deviation, 75, 76 Fitting invariant, 21{23 of maximal ideal, 84 ag variety, ASL property of coordinate ring of, 311 form, 28 initial, 186 F -pure ring, 406 fractionary ideal, 119 F -rational ring, 393 and Cohen{Macaulay property, 394 and completion, 400 and at extensions, 398 and localization, 395, 397 and pseudo-rationality, 404 Gorenstein, 395 graded, 398 is normal, 394 modulo regular element, 397, 401 F -rational type, 405 F -regular ring, 385{386 see also F -rational ring Frobenius functor, 327{331 atness of, 330 Frobenius homomorphism, 327 action on local cohomology, 400{402 Frobenius power, 328 F -stable, 402 full subsemigroup, 263 function of polynomial type, 148 f -vector, 209 and h-vector, 213 Gauss' lemma, 315 general linear form, 163 generic atness, 297 generically complete intersection, 200 generically Gorenstein, 117, 179
447 geometric realization, 225 G-graded ring, 210 going-down, 414, 415 going-up, 414, 415 Gordan's lemma, 259 Gorenstein complex, 243{246 and Euler complex, 244 simplicial sphere is, 246 topological characterization of, 244 Gorenstein ideal, 120 of grade three, 121 Gorenstein ring, 95{97, 102, 112, 118, 119 and at extension, 116 and associated graded ring, 186 and faithfully at extension, 120 de ned by monomials, 181 extremal, 155 graded, 143 is Cohen{Macaulay, 102 localization of, 95 modulo regular sequence, 95 of dimension one, 119 of dimension zero, 107 ring of type 1 is, 374 type of a local, 102 Gotzmann space, 173 Gotzmann's theorem persistence, 172 regularity, 169 Gr#obner basis, 158 grade, 8{16, 27, 36 and acyclicity, 23, 351 and dimension, 58 and exact sequence, 11 and Ext, 10 and height, 13, 59 and Koszul complex, 50{52 formulas for, 11 of a module, 12, 25, 366 of a non- nite module, 348{354 graded algebra, 28 graded ASL, 301 Cohen{Macaulay property of, 305 discrete, 304
448
Index
graded Hodge algebra, 300 Cohen{Macaulay property of, 304 discrete, 301 Gorenstein property of, 304 graded ideal, 28 graded module, 27{39 category of, 28, 38 depth of, 33, 36 dimension of, 32, 39 grade of, 36 projective, 36 projective dimension of, 36 support of, 30 type of, 33 graded ring, 27{39 Noetherian, 29 polynomial ring as, 28 trivial, 29 graded submodule, 28 Grassmannian, 306 ASL property of coordinate ring of, 307 Grothendieck's condition (CMU), 62 G-subspace, 292 g-theorem, 255 height, 412, 413 Henselian local ring, 339 Henselization, 340 Hilbert function, 147 of a positively graded algebra, 174 and f -vector, 212 and graded free resolution, 153 higher iterated, 150 of !k ], 246 of a Zn -graded module, 211 of a Gorenstein complex, 246 of a graded Gorenstein ring, 177 of a homogeneous complete intersection, 181 of a homogeneous ring, 157 of a one dimensional homogeneous domain, 181 of an Euler complex, 246 of the canonical module, 176, 177 Hilbert polynomial, 149 coe cients of, 151 Hilbert quasi-polynomial, 175
Hilbert ring, 417 Hilbert series, 147 see also Hilbert function Hilbert's Nullstellensatz, 417 Hilbert's syzygy theorem, 69 Hilbert{Burch theorem, 26 Hilbert{Samuel function, 188 Hilbert{Samuel polynomial, 188 Hochster's niteness theorem, 336{342 Hochster{Roberts theorem, 292, 386 Hodge algebra, see graded Hodge algebra homogeneous algebra, ring, 29, 147 component, 28 element, 28 homomorphism, 28 of degree i, 33 localization, 31 system of parameters, 35, 37 homogeneous socle, 142 homogenization, 39 homological height theorem, 363 h-vector, 212{214 of a homogeneous Cohen{Macaulay algebra, 166 of a level ring, 179 of an integral polytope, 277 hyperplane, 223 hypersurface ring, 298 I -adic ltration, 182
ideal generated by monomials, 181 ideal in a semigroup, 257 primary, 257 prime, 257 radical, 257 radical of, 257 ideal of de nition, 188 ideal of minors, 21{23 ideal of the principal class, 59 incidence function, 264 indecomposable module, 100 indiscrete part, 302 injective dimension, 92{95 and localization, 92 nite, 94, 119, 374, 375 modulo regular sequence, 94
449
Index
injective dimension, 138 injective hull, 97{101, 103, 107 localization of, 99 injective hull, 136 injective module, 88{92 decomposition of, 100 direct sum of, 90 localization of, 90 injective module, 136{139 injective resolution, 91, 96 minimal, 99, 101 injective resolution, 138 integral closure of an ideal, 388 of monomials, 393 integral dependence on an ideal, 387{391 and on a ring, 393 valuative criterion for, 390 integral extension, 414{415 integral over an ideal, see integral dependence on an ideal integral polytope, 276 intersection multiplicity, 366 intersection theorem, 361{367 improved new, 362, 366, 371 new, 329, 362, 366 Peskine{Szpiro, 365 Serre's, 364, 367 invariant of a group, 278
join, 221 Koszul algebra, 83 and the canonical module, 127 of a complete intersection, 78{85 of a Gorenstein ring, 125, 127 Koszul cohomology, 45, 131 Koszul complex, 43{54, 74, 75, 85 algebra structure on, 44 and at extension, 46 and grade, 50{52 as an invariant, 52 functorial properties, 48 of a sequence, 46 of regular sequence, 50 rigidity of, 54
Koszul homology, 44{54, 348, 368 algebra structure on, 44 and cohomology, 48 and at extension, 46 annihilator of, 326 exact sequence of, 49 functorial properties of, 47 Krull dimension, see dimension Krull's principal ideal theorem, see principal ideal theorem leading monomial, 157 lemma of Rees, 94 lemme d'acyclicite, 27 length of a module and at extension, 15 length of a poset, 215 level ring, 179, 240 lexsegment, 158 ideal, 161 line segment, 223 line shelling, 228 linear algebraic group, 278 linear resolution, 169 link, 232 local cohomology, 127{136, 372 and depth, 132 and dimension, 132 annihilator of, 323{327 canonical element in, 357{361 functors, 128 of a Gorenstein ring, 129 vanishing of, 132 local duality, 133, 134 for graded rings, 141{144 local ring, 35{37, 64, 72, 85 locally free module, 136 locally upper semimodular, 215 lying over, 414
Macaulay coe cient, 160 representation, 160 Macaulay's theorem on Hilbert functions, 162 on order ideals of monomials, 156 Matlis duality, 103{107 for graded modules, 142
450 maximal, 35 minimal complex of length s, 353 free resolution, 16 graded, 36 multiplicity, 191 number of generators, 16 presentation of a local ring, 74 reduction, 190 system of generators, 16 homogeneous, 36 m-linear resolution, 155 modi cation, 331{336 non-degenerate, 331 of type s, (s1 . . . sr ), 331 module of semi-invariants, 279 Molien series, 283 Molien's formula, 284 monomial theorem, 354 monomial with exponent, 300 support of, 300 multiplicity, 150, 277 and Cohen{Macaulay module, 197, 199 and rank of a module, 197 of a (non-graded) module, 188 symbol, 193 system, 192
Nagata's theorem, 315 Nakayama's lemma, graded variant, 39 neighbourly, 227 Noether normalization, 416 graded, 37 in complete local ring, 419 normal domain, see normal ring normal ring, 71{72, 117 as direct summand, 357 ring of invariants is, 279 normal semigroup, 290 normal semigroup ring, 257{261, 270{278 canonical module of, 272, 273 Cohen{Macaulay property of, 272 Gorenstein property of, 274 local cohomology of, 271 normal vector, 223
Index
normality criterion, 71, 73, 187 numerical semigroup, 178 conductor of, 178 symmetric, 178 octahedron, 208 order complex, 208 order ideal of a module element, 360, 368 order ideal of monomials, 156 orientation of a module, 47 oriented simplicial complex, 229 perfect ideal, see perfect module perfect module, 25{27, 366 Cohen{Macaulay property of, 59, 68, 69 Pfa an, 120 ideal, 318 Pl#ucker relation, 307 Poincare algebra, 125 duality, 124{127 series, 87 polyhedral set, 223 polyhedron, 223 f -vector of, 224 combinatorial equivalent, 224 dimension of, 224 edge of, 224 face of, 224 facet of, 224 subfacet of, 224 vertex of, 224 polynomial ring ASL property of, 310 polynomial ring over a eld, 69, 72 polynomials with coe cients in a module, 5 polytope, 223, 224 f -vector of, 229 see also polyhedron simplicial, 225
Index
poset, 208 bounded, 215 graded, 215 locally upper semimodular, 305 of minors, 306 pure, 215 shellable, 215 positive cone, 262 positive semigroup, 259 positively graded algebra, ring, 29, 37, 287 Noetherian, 29 prime avoidance, 8, 34 principal ideal domain, 96 principal ideal theorem, 359, 364, 412 projective dimension, 16{18, 25 nite, 66, 69, 119, 365, 367, 375 projective module, 20 projective resolution, 16 pseudo-rational ring, 393, 403 and rational singularities, 402{405 pseudo-re exion, 285 pure extension, 293 subring, 293 pure resolution, 153 pure subring, 406 quasi-polynomial, 174 quasi-regular sequence, 5{8, 52, 342 rank of a homomorphism, see rank of a module of a module, 19{21, 27, 54 of a poset element, 215 of a semigroup, 257 with respect to a module, 351 rational cone, 258 half-space, 258 polytope, 258 real projective plane, 236 reduced simplicial (co)homology, 230 relative, 241
451 reduced singular homology and simplicial homology, 231 and cell complex, 266 of a sphere, 231 reduction ideal, 189, 388 reduction to characteristic p, 65, 296, 336, 363 Rees ring, 182 and reduction ideal, 189 dimension of, 185 extended, 182 minimal prime ideals of, 184 re exive module, 19, 27, 107 regular element, 3, 9, 21 regular equational presentation, 337 regular local ring, 65{73, 97 as a direct summand, 354{357 associated graded ring, 66 Cohen{Macaulay property of, 66 completion of, 65 factoriality of, 70 localization of, 67 normality of, 71 of characteristic p, 330{331 regular ring, 68{73, 96, 97 and ground eld extension, 70 and polynomial extension, 69 and power series extension, 69 graded, 72 positively graded, 72 regular sequence, 3{8, 15, 67, 79, 342 and completion, 4 and nite projective dimension, 365 and at extension, 4 and localization, 4 and system of parameters, 12 homogeneous, 34 of 1-forms, 35 regular system of parameters, 65, 79 regularity (Castelnuovo{Mumford), 168 Reisner's criterion, 235 relative interior, 225, 261 representation of a group, 278 resolution with algebra structure, 82 reverse degree-lexicographic order, 156
452 Reynolds operator, 281, 282, 292 ring of characteristic p, 327 ring of invariants, 278{298 a-invariant of, 296 and regularity, 287 canonical module of, 280, 285, 296 Cohen{Macaulay property of, 280, 283, 292 complete intersection property of, 290 Gorenstein property of, 281, 285, 296 of a diagonalizable group, 279{281 of a nite group, 281{291 of a group of pseudo-re exions, 286{291 of a linearly reductive group, 292{297 ring of type one, 374 saturation, 168 second deviation, 81 self-dual complex, 48 semi-invariant of a group, 279 separable, 70 sequence, see regular sequence Serre's condition (Rn ), 71, 73 Serre's condition (Sn ), 63{64, 71, 73 Serre's normality criterion, 71 shellable complex, 214{221 h-vector of, 218 is Cohen{Macaulay, 217 shelling, 214 of a polytope, 228 simplex, 208, 225 simplicial complex, 207 acyclic, 230 connected, 222 constructible, 219 core of, 244 dual, 241 of a simplicial polytope, 225 pure, 210 restriction of, 241 shellable, see shellable complex simplicial sphere, 237 singular locus, 382 skeleton, 221
Index
small support of a module, 345 socle, 15, 79 solution of height n, 336 split acyclic complex, 23 standard basis, 158 standard monomial, 301 standard representation, 302 Stanley's reciprocity law, 275 Stanley{Reisner ring, 209, 301 and punctured spectrum, 237 canonical module of, 246{254 Cohen{Macaulay property of, see Cohen{Macaulay complex dimension of, 210 Gorenstein property of, see Gorenstein complex homogeneous system of parameters of, 220 local cohomology of, 232{236 minimal prime ideals of, 210 of order complex, 304 Stanley{Reisner rings Betti numbers of, 240{243 star, 232 straightening law, relation, 301 on poset of minors, 306{311 super cial element, 192 superheight, 363 support of a monomial, 211, 300 of an element a 2 Zn , 211 supporting hyperplane, 224 symmetric ASL, 311 system of homogeneous linear diophantine equations, 274 system of parameters, 334, 336, 354, 413 and multiplicity, 195 syzygy, 16, 64 rank of, 367{371 test element, 398{399 tight closure, 378 and completion, 381, 399 and integral closure, 391{393 and localization, 381, 396 and module- nite extension, 381 and regular rings, 383
453
Index
and system of parameters, 384 of -primary ideal, 385 of a principal ideal, 392 tightly closed ideal, 378 primary component of, 387 torsion module, 19 torsion-free module, 19, 26, 27 torsionless module, 19, 27 torus, 279 transpose of a module, 27 transversal, 262 triangulation, 231 trivial extension, 118 type of a module, 13{15, 102, 103, 114 and at extension, 13 and localization, 115 UFD, see factorial ring universally catenary, 62 unmixed ideal, 59, 63 unmixed ring, 63 m
upper bound theorem for polytopes, 227 for simplicial complexes, 237{240 valuation ring, 390 vanishing of ExtiR (M !R ), 135 Veronese subring, 144 vertex, 207 scheme, 225 visible, 270 volume of a polytope, 277 weak sequence, see weakly regular sequence weakly F -regular, see F -regular ring weakly regular sequence, 3{9, 50, 353 -regular module, 3
x
Zn -grading, 210
